Categories of abelian varieties over finite fields II: Abelian varieties over finite fields and Morita equivalence

The category of abelian varieties over $\mathbb{F}_q$ is shown to be anti-equivalent to a category of $\mathbb{Z}$-lattices that are modules for a non-commutative pro-ring of endomorphisms of a suitably chosen direct system of abelian varieties over $\mathbb{F}_q$. On full subcategories cut out by a finite set $w$ of conjugacy classes of Weil $q$-numbers, the anti-equivalence is represented by what we call $w$-locally projective abelian varieties.


Introduction
1.1.The scene.Let q " p r be a power of a prime number p, and let F q be a finite field with q elements.In this paper we generalize [CS15] to the category AV Fq of abelian varieties over F q .Our main result says that there is a non-commutative pro-ring S q and an anti-equivalence T : AV Fq ÝÑ Mod Z-tf pS q q (1.1) between AV Fq and the category of left S q -modules that are free and of finite rank over Z.The ring S q arises in the construction of (1.1) and admits a description as a pro-ring for Z-orders S w of certain finite Q-algebras.The construction of T involves several choices.As a consequence S q is not unique, however, its center R q is unique and can be described explicitly in terms of the set of Weil q-numbers.Besides having its own interest, our result provides a non-commutative algebra into which potentially every question on AV Fq can be translated.It will be clear to the reader that our work is permeated by Honda-Tate theory [Ho68, Ta68] and by Tate's result on the local structure of the Hom-groups [Ta66].Since these milestones for the subject were placed in the 1960s, variants of the anti-equivalence (1.1) for several subcategories of AV Fq have been studied by many mathematicians, with reviving interest in recent years.We recall results on this classical topic in a similar spirit as ours, and apologize in advance for possible omissions.Waterhouse [Wa69] studied isomorphism classes and endomorphism rings of Date: May 9, 2022.The second author acknowledges support by Deutsche Forschungsgemeinschaft (DFG) through the Collaborative Research Centre TRR 326 "Geometry and Arithmetic of Uniformized Structures", project number 444845124.
certain simple objects of AV Fq .More recently, Yu [Yu12, Theorem 3.1] classified isomorphism classes of objects of AV Fp whose characteristic polynomial of Frobenius is relatively prime to x 2 ´p.Giraud [Gi68,§1] and Waterhouse [Wa69,Appendix] (see also [JKP+18,§4]) exploited a functor due to Serre and Tate in the converse direction as (1.1) to study elliptic curves.Deligne [De69] used the canonical lifting of Serre and Tate to give a complete description of the full subcategory of AV Fq consisting of ordinary objects.This construction was further developed by Howe [Ho95] to solve questions concerning polarizations, a topic also addressed in a similar fashion by Bergström, Karemaker and Marseglia [BKM21].The canonical lifting technique was extended by Oswal and Shankar [OS20] and used to classify objects in isogeny classes of simple, almost ordinary abelian varieties.Classification of abelian varieties isogenous to powers of a given simple abelian variety (often an elliptic curve) have been the focus of attention in [Ka11], [Yu12], [JKP+18].
Our emphasis is on a uniform categorical description of all of AV Fq in terms of modules.
1.2.Frobenius and Weil numbers.Before giving more details on our method and stating a precise theorem we recall some notation and terminology.Two Weil q-numbers π and π 1 are conjugate if there is an isomorphism Qpπq » Qpπ 1 q sending π to π 1 .The set of conjugacy classes of Weil q-numbers is denoted by W q .When no confusion is likely to arise, we may suppress the distinction between a Weil number and its conjugacy class.For any π P W q , we fix once and for all an F q -simple abelian variety B π belonging to the isogeny class of AV Fq defined by π according to Honda-Tate theory [Ho68,Ta68].
The Weil support wpAq of an object A of AV Fq is the finite subset of W q consisting of Weil numbers associated to the simple factors of A. This is to say that there is an F q -isogeny with positive multiplicities n π .For a subset w Ď W q , finite or infinite, we denote by AV w the full subcategory of AV Fq whose objects are those abelian varieties A such that wpAq Ď w.
For w Ď W q finite, the minimal central order R w is the largest quotient through which the ring homomorphism ZrF, V s{pF V ´qq ÝÑ Qpwq :" sending F and V to the diagonal images of π and q{π respectively, factors.As w ranges through the finite subsets of W q , the R w form a pro-ring R q " pR w , r w,w 1 q, where the transition maps r w,w 1 : R w 1 Ñ R w are the natural surjections, defined when w Ď w 1 .The category AV w has a natural R w -linear structure which varies compatibly as w increases.This datum forms what we refer to as the R q -linear structure on AV Fq .
1.3.The main result.The strategy for constructing the anti-equivalence (1.1) is similar to that followed in [CS15].We first construct, for any finite subset w Ď W q , what we call a w-balanced abelian variety A w , see Definition 2.8.We then show that A w represents an anti-equivalence T w : AV w ÝÑ Mod Z-tf pS w q, T w pXq " Hom Fq pX, A w q, (1.3) where S w " End Fq pA w q and Mod Z-tf pS w q is the category of left S w -modules that are free and of finite rank over Z.The w-balanced object A w and hence S w is not unique.However the center of S w is isomorphic to R w under the map induced by the R w -linear structure of AV w .The second step is to show that the objects A w can be chosen compatibly as w increases.More precisely, we prove that there is a family of w-balanced abelian varieties A w and an ind-object A " pA w , ϕ w,w 1 q such that the corresponding ind-representable functor Hom Fq p´, Aq " lim Ý Ñ w Hom Fq p´, A w q interpolates the anti-equivalences (1.3) for each w.An important by-product of the construction of A is that the endomorphism rings S w form a projective system S q " End Fq pAq " pS w , s w,w 1 q with r w,w 1 -linear and surjective transition maps s w,w 1 : S w 1 Ñ S w , for w Ď w 1 .Denote by Mod Z-tf pS q q the category of S q -modules for which the structural action factors through some S w , and which as Z-modules are free of finite rank.This category is R q -linear and for each object there is a clear notion of support, parallel to that in AV q .Our main result can be formulated as follows.
Theorem 1.1.Let q " p r be a power of a prime number p.There exists an ind-abelian variety A " pA w , ϕ w,w 1 q such that A w is w-balanced for all finite w Ď W q and the transition maps are inclusions.With S q " End Fq pAq, the contravariant and R q -linear functor T : AV Fq ÝÑ Mod Z-tf pS q q, T pXq " Hom Fq pX, Aq is an anti-equivalence of categories which preserves the support.Moreover, the Z-rank of T pXq is equal to 4r dimpXq.
Remark 1.2.(1) The main difference with [CS15] is that the varieties A w can no longer be chosen to be multiplicity free in general (see Theorem 3.10), and so their endomorphism rings S w are non-commutative, hence harder to describe explicitly, see Theorem 4.9.We consider the use of multiplicities as a major insight which is stimulated by and used here in the context of Morita equivalence.This explains how we decided the title of this note.
(2) The rank of the associated Z-lattice T pXq exceeds the dimension of H 1 pXq in any Weil cohomology by a factor of 2r.This cannot be avoided in general, see Proposition 8.9.However the ratio rk Z pT pXqq{2 dimpXq can be lowered to r if r is even or by excluding real Weil numbers from the scope of the anti-equivalence, see Remark 7.2.
(3) The rational version of Theorem 1.1 follows easily from Honda-Tate theory.Here rational means ´b Q for modules and working up to isogeny for abelian varieties.In fact, Theorem 1.1 can be seen as a categorification of an integral version of Honda-Tate theory.
1.4.Geometry of the category of abelian varieties.Let w be a finite set of Weil q-numbers.
Restricted to AV w , our main result has a geometric interpretation on the 1-dimensional scheme X w " SpecpR w q.The R w -algebra S w corresponds to a coherent sheaf of (in general) noncommutative O Xw -algebras S w .Finitely generated modules for S w are coherent sheaves on X w endowed with an O Xw -linear S w -module structure.We call these coherent S w -modules.The anti-equivalence (1.3) describes abelian varieties from AV w as coherents sheaves of S w -modules that are flat over SpecpZq.
With this point of view, it makes sense to talk about categorically local properties of an abelian variety X in AV w at a prime " p (or at p) as properties of the -adic (or p-adic) completion of the module Hom Fq pX, A w q.With hindsight, one should be able to detect local structure of X in a more direct way.Indeed, Tate's theorems identify local structure of X in terms of the Galois module T pXq (resp.the Dieudonné module T p pXq).The proof of our main result compares the datum of the Tate module and the completion of Hom Fq pX, A w q based on the special properties of w-balanced objects.We therefore adopt the terminology of local properties of abelian varieties if these are defined in terms of Tate modules only (see Definition 3.1 for a refinement which indicates that the present point of view is only semi-local).
The following definition is an important example of a local property.We use the notation D w to denote a certain quotient of the Dieudonné ring to be explained in (2.5) of Section 2.1.Definition 1.3.Let q " p r be a power of a prime number p, and let w Ď W q be a finite subset of Weil q-numbers.An abelian variety A P AV w is w-locally projective if (i) for all " p the Tate module T pAq is a projective R w b Z -module, and (ii) T p pAq is a projective D w -module.
The w-balanced abelian varieties, constructed in Theorem 2.7, are w-locally projective.The w-locally projective objects of AV w can be characterized as follows.
Theorem 1.4.Let w Ď W q be a finite subset, and let A in AV w be an abelian variety with S " End Fq pAq.Then the following are equivalent.(a) A is w-locally projective with support wpAq " w.

(b)
The functor Hom Fq p´, Aq : AV w ÝÑ Mod Z-tf pSq is an anti-equivalence of categories.
This result will follow from Theorem 6.6.More on the classification of w-locally projective abelian varieties is explained by Theorem 3.10 and Theorem 8.5.
Remark 1.5.While w-locally projective is the decisive property for the representable antiequivalence of AV w with a suitable category of modules (see Theorem 6.6), we have formulated our main result Theorem 1.1 in the more restrictive setting using w-balanced abelian varieties.The w-balanced abelian varieties are easy-to-construct (see Theorem 2.7) examples of w-locally projective objects which fit well together to form an ind-object suitable to prove our main result, see Section §7.2.Furthermore, (reduced) w-balanced abelian varieties enjoy a minimality property among the larger set of w-locally projective ones, as long as w is large enough and SpecpR w q is connected (see Theorem 3.10).
1.5.The commutative case.Our method can be applied also to the study of any full subcategory of AV Fq of the form AV W , where W Ď W q is any subset.The outcome is the existence of a pro-ring S W and an ind-representable anti-equivalence between AV W and the category Mod Z-tf pS W q of modules over S W that are finite and torsion free over Z.As in the case where W " W q , the functor T W and the pro-ring S W appearing in (1.4) are far from being unique.The center of any such S W is the quotient R W of R q defined by W .We say that a subset W Ď W q is a commutative set of Weil numbers if there is a functor T W p´q as in (1.4) for which S W is commutative.There are two examples: ‚ the subset W ord q of ordinary Weil q-numbers and, ‚ for q " p, the set W com p " W p zt˘?pu given by the complement of the real conjugacy class of Weil p-numbers, see [CS15] and Section §7.2.The existence of (1.4) for W " W ord q was first shown by Deligne in [De69] (up to switching the variance of T W ), who exploited the existence of the Serre-Tate canonical lifting for ordinary objects.In §8.1, see Theorem 8.4, we reprove his result without involving lifts to characteristic zero.Instead, our argument relies on a calculation of an endomorphism ring.Then a Moritaequivalence trick deduces the claimed anti-equivalence of categories.
These two cases W ord q and W com p are the only maximal commutative W , see Proposition 8.8, and in this case the anti-equivalence (1.4) reads as with the rank of T W pXq being equal to 2 dimpXq.
1.6.Outline.This article is structured as follows.In Section §2 we recall Honda-Tate theory which can be considered as the rational version of our anti-equivalence of categories.This leads directly to the existence of w-balanced abelian varieties in Theorem 2.7, a choice of integral structure within the rational theory.The study of the local integral theory starts in Section §3 with the definition of the local Tate modules T λ pAq (resp.T p pAq) for maximal ideals λ (resp.p) of R w .We moreover compute in Section §4 in Theorem 4.9 a description of the endomorphism ring of balanced abelian varieties and thus make the main theorem more concrete and potentially accessible to computations.
In Section §5 we provide a formal criterion (Morita equivalence, Theorem 5.4) for a representable functor (in our setting) to be an anti-equivalence.We subsequently apply the criterion in Section §6 to w-locally projective abelian varieties, in particular to w-balanced abelian varieties: these are injective cogenerators for the respective w-truncated category of abelian varieties AV w .It should be emphasized that this step relies on Tate's theorems (2.4) and (2.7), and, for the prime-to-p part, on the minimal central order R w being Gorenstein.We moreover determine the center of S q in Section 6.3.Section §7 glues the truncated anti-equivalences by a limit argument and so completes the proof of Theorem 1.1.
Section §8 adresses special subcategories for which the multiplicity of the representing object can be lowered, and we show in Proposition 8.9 that multiplicities are unavoidable in general.
We conclude in Section §9 with an instructive example that shows why multiplicity for the injective cogenerator is important (there are two non-isomorphic simple objects) and that the cogenerator is not just a product of simple objects but involves a congruence (the product has a cyclic isogeny onto the injective cogenerator).Moreover, we classify isogeny classes of injective cogenerators for Weil numbers of supersingular elliptic curves.
Acknowledgements.We are grateful to the anonymous referee for numerous suggestions that helped improve the presentation of our results.

Rational theory and choice of a lattice
We recall Honda-Tate theory in order to fix notation and to prepare for Theorem 2.7.
2.1.Minimal central orders.For a Weil q-number π, denote by P π pxq its monic minimal polynomial over Q, and by h π pF, V q the associated symmetric polynomial, which is uniquely determined by the equation h π px, q{xq " x ´degpPπq{2 ¨Pπ pxq and being a linear combination of powers F i and V j with i, j P 1 2 Z and i, j ě 0. The symmetric polynomial h π pF, V q lies in the polynomial ring ZrF, V s if π is not rational, as then degpP π q is even.If π is rational, then r is even and π " ε ?q with ε P t1, ´1u and the positive rational square root ?q.The symmetric polynomial then is h π pF, V q " F 1{2 ´εV 1{2 and lies in ZrF 1{2 , V 1{2 s.For a finite subset w Ď W q , we define P w pxq " We moreover define the degree of w as degpwq " degpP w q " rQpwq : Qs " The degree degpwq is even unless r " rF q : F p s is even and w contains exactly one of the two rational Weil q-numbers ?q and ´?q.
Let us recall the structure of the minimal central order R w from [CS15, §2.5].If w either contains both or no rational Weil q-numbers, i.e. if degpwq " 2d is even, then the natural map induces an isomorphism ZrF, V s{pF V ´q, h w pF, V qq " ÝÑ R w , and shows that we have a Z-baisis of R w by the elements represented by F d , . . ., F, 1, V, . . ., V d´1 or alternatively by F d´1 , . . ., F, 1, V, . . ., V d . (2.1) If, on the other hand, we have w equal to w 0 Y tε ¨pm u, where r " 2m is even, ε P t1, ´1u, and w 0 contains no rational Weil q-number, then the degree degpwq " 2d 0 `1 is odd.Moreover, we have an isomorphism and a Z-basis of R w is given by the elements represented by (2.2) 2.2.Localizations of AV Fq .Let be a prime number different from p.As usual, for an object A of AV Fq we denote the -adic Tate module of A by T pAq, and by V pAq the Q -vector space T pAq b Z Q .For any finite subset w Ď W q , and any A in AV w , the R w -linear structure on AV w induces on T pAq the structure of an R w b Z -module.The ring R w b Z is generated over Z by image of F , since ‰ p, and we have Let F p be a fixed algebraic closure of the prime field F p , and consider F q as a subfield of F p .The Galois action on T pAq can be recovered from the R w bZ -structure, in that F acts on T pAq as the arithmetic Frobenius Frob q P GalpF p {F q q.In particular, maps as Galois representations between -adic Tate modules for objects in AV w are the same as R w b Z -module homomorphisms.The theorem of Tate [Ta66, Main Theorem] thus says that the functor T p´q induces an isomorphism To study the localization of AV Fq at p, consider the Dieudonné ring Here W pF q q is the ring of Witt vectors of F q , the variable F is σ-linear and V is σ ´1-linear where σ is the arithmetic Frobenius of W pF q q{Z p .Moreover, F and V commute with each other.We introduce the notation The ring D q is a free W pF q q-module with basis pF i q iPZ , and is an isomorphism onto the center of D q .The contravariant Dieudonné module T p pAq of an object A of AV Fq is a left D q -module which is free of rank 2 dimpAq as a W pF q q-module.The Q p -vector space T p pAq b Zp Q p will be denoted by V p pAq.For any finite subset w Ď W q and any A in AV w , just like in the ‰ p case, we deduce an R w b Z p -structure on T p pAq such that F acts on T p pAq as the central element F r P D q and V as V r .Let degpwq " 2d be even, i.e. w contains either both rational Weil q-numbers or none.Then D q acts on T p pAq via its quotient which is a free W pF q q-module with basis by the elements represented by F ´dr , . . ., F dr´1 or alternatively F ´dr`1 , . . ., F dr .
Let now degpwq " 2d 0 `1 be odd, i.e., w is equal to w 0 Y tεp m u, where r " 2m is even, ε P t1, ´1u, and w 0 contains no rational Weil q-number.Then D q acts on T p pAq via its quotient D w :" W pF q qtF , V u{ pF V ´p, h w 0 pF r , V r qpF m ´εV m qq which is a free W pF q q-module with basis by the elements represented by F ´p2d 0 `1qm , . . ., F p2d 0 `1qm´1 or alternatively F ´p2d 0 `1qm`1 , . . ., F p2d 0 `1qm . (2.6) We remark that pF m ´εV m q ¨F m " F ´?q and similarly pF m ´εV m q ¨V m " ´εpV ´?qq.
In both cases, even and odd degree degpwq, we find a natural map Proof.Injectivity follows by comparing the Z p -basis of R w b Z p (arising from the Z-basis of R w described in (2.1) and (2.2)) with the W pF q q-basis of D w described in (2.6).The image is also clearly contained in the center since the r-th power of the Frobeniuis σ of W pF q q equals the identity.
The number N " degpwqr{2 is an integer and F i for ´N ď i ă N form a W pF q q-basis of D w .Let x " ř N ´1 i"´N x i F i be an element of the center of D w .For an arbitrary a P W pF q q we have It follows that x i " 0 for all i not divisible by r.Next, let m i " 1 for i ă 0 and m i " 0 for i ě 0. Then the equation F x " xF yields Since the F i for ´N ă i ď N also form a W pF q q-basis of D w , we find that all x i are in Z p .Hence x lies in the image of R w b Z p Ñ D w .
Using the language of Dieudonné modules, Tate's theorem [WM71, Part II Theorem 1] says that the natural map gives an isomorphism Hom Fq pA, Bq b Z p " ÝÑ Hom Dw pT p pBq, T p pAqq (2.7) for all A, B in AV w .
2.3.Rational local module structure.For any π P W q , recall that B π is an F q -simple abelian variety associated to π via Honda-Tate theory.The ring of endomorphisms of B π in the isogeny category of abelian varieties over F q is a central division ring over the subfield Qpπq.The index of E π will be denoted by s π , so that rE π : Qpπqs " s 2 π .While E π is determined by π up to isomorphism, it is well known that its order End Fq pB π q is not an isogeny invariant in general.The structure of E π as a central simple algebra over Qpπq is determined by its local invariants.
(1) The dimension of the simple object B π is computed by the dimension formula 2 dimpB π q " s π rQpπq : Qs.
(2) The local invariant of E π " End Fq pB π q b Q at a place v of its center Qpπq is the following element of Q{Z: where f v " rK v : Q p s is the inertial degree of Qpπq at v | p.
Let now be a prime number including for the moment possibly " p.The completions K v of Qpπq at the places v | are factors of the product decomposition (2.8) The algebra E π b Q has center Qpπq b Q and (2.8) similarly induces a decomposition where E v is a central simple algebra over K v .Since the idempotents cutting out K v from Qpπq b Q lie in the center of E π b Q , we have an analogous decomposition as a module under E π b Q with the summand V v pB π q being an E v -module.For a prime ‰ p, the rational Tate module V pB π q has both an action by E π b Q and a commuting action by the absolute Galois group of F q .The latter was identified in Section 2.2 as the action of the center R π bQ of E π bQ .Hence the decomposition (2.9) is also a decomposition as Galois representations, with K v acting on V v pB π q capturing the Galois action.The summands have the following structure.
Lemma 2.3.For a prime ‰ p and a place v | of Qpπq, there is an isomorphim (2.10) as Galois modules, i.e. as modules for K v .In particular, there is an isomorphism (2.11) as Galois modules, i.e. as modules for Qpπq b Q .
Proof.From Tate's Theorem (2.4) for A " B " B π we deduce, by applying base change along The algebra E v has K v -dimension s 2 π as a base change of E π .Thus s π " dim Kv pV v pB π qq and (2.10) follows because K v is a field.The existence of the isomorphism (2.11) follows from (2.10) together with (2.8), since the exponents s π are independent of v.
We now turn our attention to the local structure at p.The rational Dieudonné ring D 0 q " W pF q qtF , V u{pF V ´pq b Zp Q p is central over the algebra Q p rF, F ´1s with F " F r .We set W pF q q 0 " W pF q qr 1 p s, and observe that W pF q q 0 b Qp Q p rF, F ´1s is a cyclic étale Q p rF, F ´1s-algebra of degree r and canonical generator σ b id ": σ of its Galois group.This imposes on D 0 q the structure of a cyclic Azumaya algebra D 0 q " `W pF q q 0 b Qp Q p rF, F ´1s, σ, F ȏver Q p rF, F ´1s.The action of D 0 q on V p pB π q factors through the base change with namely the quotient D 0 π " D 0 q b QprF,F ´1s pQpπq b Q p q " W pF q q 0 tF u{pP π pF r qq.
The Q p -linear map sending π to F " F r gives an isomorphism between Qpπq b Q p and the center of D 0 π .As a base change of a cyclic algebra, the ring D 0 π is itself a cyclic algebra over Qpπq b Q p as follows.Consider as a cyclic étale Qpπq b Q p -algebra of degree r and σ b id " σ as a distinguished generator of the Galois group.Then D 0 π " pL π , σ, πq " L π tF u{pF is σ-semilinear, and F r " πq (2.12) as a cyclic Azumaya algebra of index r over Qpπq b Q p .After decomposing D 0 π according to the components K v of its center, we obtain a decomposition into the product over the p-adic places of Q p of certain central simple algebras D 0 π,v over K v , each of index r.
The rational Dieudonné module V p pB π q is simultaneously a left module for the Dieudonné ring D 0 π and for the opposite endomorphisms actions by E op π bQ p .Since the idempotents cutting out K v commute with both these actions, we can decompose as a module for both actions.The summands have the following structure.
Lemma 2.4.For a place v | p of Qpπq, there is an isomorphism (2.13) as Dieudonné modules, i.e. as modules for D 0 π,v .In particular, there is an isomorphism as Dieudonné modules, i.e. as modules for D 0 π .
Proof.From Tate's Theorem (2.7) for A " B " B π we deduce, by applying base change along This means that E op v is the centralizer of D 0 π,v in End Kv `Vv pB π q ˘.A corollary to the double centralizer theorem, [Re75, Corollary 7.14], then states that In particular, comparing the degrees of these simple central K v -algebras, we find s π ¨r " dim Kv pV v pB π qq.
Since modules over D 0 π,v are determined, up to isomorphism, by their K v -dimension, assertion (2.13) follows by comparing dimensions dim Kv `Vv pB π q 'r ˘" r ¨dim Kv pV v pB π qq " s π ¨r2 " dim Kv `pD 0 π,v q 'sπ ˘.
The existence of the isomorphism (2.14) follows from (2.13) together with (2.8), since the exponents s π and r are independent of v.
2.4.Choosing good multiplicities and choice of a lattice.The index s π of E π agrees with the period of E π , defined to be the order of E π in the Brauer group BrpQpπqq of Qpπq (essentially due to [BHN32], see [Re75] Theorem 32.19 for a textbook reference).By the Hasse-Brauer-Noether Theorem the period of E π is the least common multiple over all places of Qpπq of all local orders of E π in the appropriate Brauer groups.Since the localization of E π at any -adic place is trivial for ‰ p, only p-adic and real places (if any) of Qpπq must be taken into account to compute the global order s π from the local ones.
The denominator of the local invariant of E π at a p-adic place p divides r by Theorem 2.2 (2).Because the local invariant of E π at a real place of Qpπq equals 1{2, we conclude that s π | lcmpr, 2q, and in particular s π divides 2r in all cases.Notice that unless r is odd and π is the real conjugacy class t˘?qu of Weil q-numbers, then s π divides r.
Definition 2.5.The balanced multiplicity of π is the integer Unless r is odd and π is real we define the reduced balanced multiplicity of π to be the integer This choice of m π (resp.s m π ) ensures that the ranks are independent of π P W q in the following freeness statement for the rational Tate and Dieudonné modules of B mπ π (resp.B s mπ π ).Proposition 2.6.Let π be a Weil q-number.There is an isomorphism (1) V pB mπ π q » pQpπq b Q q '2r of Qpπq b Q -modules, for any prime ‰ p, and (2) V p pB mπ π q » pD 0 π q '2 of D 0 π -modules.Unless r is odd and π " ˘?q there is an isomorphism (3) V pB s mπ π q » pQpπq b Q q 'r of Qpπq b Q -modules, for any prime ‰ p, and (4) V p pB s mπ π q » D 0 π of D 0 π -modules.Proof.The proposition follows from the fact that V pB π q is a product of modules over simple algebras, and from Lemmas 2.3 and 2.4.Proposition 2.6 admits the following integral refinement.
Theorem 2.7.Let w Ď W q be a finite subset.There is an abelian variety A w , isogenous to ś πPw B mπ π , such that (1) T pA w q » pR w b Z q '2r as Galois modules, for all primes ‰ p.
(2) T p pA w q » pD w q '2 as Dieudonné module.
If r is even or w avoids π " ˘?q there is an abelian variety s A w , isogenous to (3) T p s A w q » pR w b Z q 'r as Galois modules, for all primes ‰ p. (4) T p p s A w q » D w as Dieudonné module.
Proof.We first work rationally and set A 1 w " ś πPw B mπ π .Proposition 2.6 thus gives isomorphisms (1) V pA 1 w q » pQpwq b Q q '2r , for any ‰ p, (2) V p pA 1 w q » pD 0 w q '2 .Both sides of these isomorphisms contain natural lattices: T pA 1 w q versus pR w b Z q '2r , and T p pA 1 w q versus pD w q '2 .For abstract algebraic reasons we may modify all but finitely many of the isomorphisms forcing them to respect these lattices.The point is that for suitable N P Z the localized ring R w r 1 N s is a product of Dedekind rings, hence after completing at , for N p, all torsion free modules are locally free.Since the rank of T pA 1 w q is constant and equal to 2r we may even deduce that it is necessarily free.A suitable modification of the isomorphism then maps one free lattice into the other.
It remains to deal with finitely many primes , and the prime p.But here the two lattices on both sides are commensurable, hence without loss of generality we can assume that one is contained in the other, after rescaling the isomorphisms.Then a suitable isogeny allows us to replace A 1 w with the required A w .The construction of s A w with reduced balanced multiplicities follows with the same proof.
Definition 2.8.Let A be an abelian variety in AV w .
(1) We say that A is w-balanced if A satisfies the conditions of Theorem 2.7 with respect to the balanced multiplicities m π .An abelian variety is balanced if it is w-balanced for some finite set w of Weil numbers (necessarily with w equal to the Weil support of the abelian variety).
(2) We say that A is reduced w-balanced if A satisfies the conditions of Theorem 2.7 with respect to the reduced balanced multiplicities s m π .
A (reduced) w-balanced abelian variety is w-locally projective in the sense of Definition 1.3.

Integral local theory and locally projective abelian varieties
Let w be a finite set of Weil q-numbers.Let " p be a prime number.The ring R w b Z is a semi-local Z -algebra, all maximal ideals λ lie over p q and there is a canonical product decomposition where R w,λ " pR w bZ q λ is the localization (isomorphic to the completion of R w at the maximal ideal corresponding to λ).Similarly, we can decompose R w b Z p " ś p R w,p with respect to the maximal ideals p of R w b Z p and the localization R w,p " pR w b Z p q p .Definition 3.1.Let A be an abelian variety in AV w .Let λ (resp.p) be a maximal ideal of R w b Z (resp. of R w b Z p q for some " p. (1) We define the λ-adic Tate module of A as the R w,λ -module (2) We define the p-component of the Dieudonné ring as and the p-adic Tate module of A as the D w,p -module The analogs of Tate's theorems (2.4) and (2.7) continue to hold.Proposition 3.2.Let A and B be abelian varieties in AV w .
(1) For all maximal ideals λ of R w b Z with " p, the functor T λ p´q induces an isomorphism (2) For all maximal ideals p of R w b Z p , the functor T p p´q induces an isomorphism Proof.This follows at once from Tate's theorems (2.4) and (2.7) in view of base change by the flat map R w b Z Ñ R w,λ (resp.the flat map R w b Z p Ñ R w,p ).Definition 3.3.Let q " p r be a power of a prime number p, and let w Ď W q be a finite subset of Weil q-numbers.An abelian variety A P AV w is w-locally free if (i) for all maximal ideals λ of R w b Z the Tate module T λ pAq is a free R w,λ -module, and (ii) for all maximal ideals p of R w b Z p the Tate module T p pAq is a free D w,p -module.
Remark 3.4.The notion of an abelian variety A in AV w being w-locally projective, see Definition 1.3, is compatible with the point of view towards local properties taken in Definition 3.3, because T pAq is projective if and only if T λ pAq is projective for all λ | and similarly T p pAq is projective if and only if T p pAq is projective for all p | p.
Lemma 3.5.Let w Ď W q be a finite set of Weil q-numbers.Let p be a maximal ideal of R w b Z p such that F and V are in p.
Let M be a finitely generated D w,p -module, and set M :" M {pF , V qM .Then the following holds.
(1) The ideal p equals pp, F, V q, and the ring D w,p {pF , V q equals F q .
(2) Elements x 1 , . . ., x m P M generate M as a D w,p -module if and only if their images generate M as an F q -vector space.
Proof.Assertion (1) follows from the definition of R w b Z p and of D w,p .The assumption implies that the constant term in h w pF, V q is divisible by p and thus pF , V q is actually a nontrivial two-sided ideal.
Assertion (2) follows from assertion (3) applied to the cokernel of the map D 'm w,p Ñ M induced by the tupel of elements x 1 , . . ., x m .It therefore remains to show that M " 0 implies M " 0.
Since R w,p is a finite Z p -module, the p-adic topology agrees with the p-adic topology.Therefore p n Ď pR w,p for sufficiently large n.By assumption then pF , V q nr Ď pF r , p, V r q n " p n D w,p Ď pD w,p Ď pF , V q.
It follows that the p-adic topology on M agrees with the pF , V q-adic topology.Since M is finitely generated as a D w,p -module, it is also finitely generated as a Z p -module and thus padically hausdoff.Therefore 0 " If M " 0, then M equals pF , V qM , and the latter intersection equals M .
Proposition 3.6.Let p be a maximal ideal of R w b Z p .Up to isomorphism there is a unique non-zero finitely generated indecomposable projective D w,p -module P w,p .All finitely generated projective D w,p -modules are isomorphic to multiples P 'm w,p for some m ě 0. If p equals the supersingular maximal ideal p o " pF, V, pq, then the indecomposable projective module P w,po is the free D w,po -module of rank 1.
Proof.There are three cases.If F R p, then F is a unit in R w,p .So V " qF ´1 holds in R w,p and D w,p " `W pF q q b R w,p ˘tF u{pF r ´F q is a cyclic algebra over the complete local ring R w,p with finite residue field.It follows from [Mi80, IV Proposition 1.4] and the fact that finite fields have trivial Brauer group that this cyclic algebra is trivial, i.e. there is an R w,p -algebra isomorphism D w,p » M r pR w,p q.
(3.1)By Morita equivalence, projective D w,p -modules are translated into projective R w,p -modules, which are free.So all projective R w,p -modules are multiples of a unique indecomposable one.This translates back by Morita equivalence to D w,p -modules.The module P w,p corresponds to the M r pR w,p q-module of column vectors pR w,p q r .If V R p, then we have F " qV ´1 in R w,p and the above holds with F and V interchanged.So again we have D w,p » M r pR w,p q (3.2) with the same conclusion.
In the third case both F and V are contained in p and we are in the situation of Lemma 3.5.Let P be a finitely generated projective D w,p -module, and let m " dim Fq P {pF , V qP .Then, by choosing an F q -basis of P {pF , V qP , Lemma 3.5 (2) shows there is a surjection This surjection splits since P is projective.Therefore Q " kerpf q is a direct summand, and consequently also finitely generated as a D w,p -module.Since f modulo pF , V q is an isomorphism, we deduce that Q{pF , V qQ " 0. Lemma 3.5 (3) implies Q " 0 and hence P » D 'm w,p is free.So in this case P w,p equals D w,p .
The connection between the notions 'w-locally projective' and 'w-locally free' is summarized by the following proposition.
Proposition 3.7.Let q " p r be a power of a prime number p, and let w Ď W q be a finite subset of Weil q-numbers.Let A be an abelian variety in AV w .
(1) If A is w-locally free, then A is w-locally projective.
(2) If A is w-locally projective, then there is an n ě 1 such that A n is w-locally free.In fact n " r always works.
Proof.Assertion (1) is obvious because free modules are projective.We now prove (2) and assume that A is w-locally projective.Tate modules are finitely generated modules, and finitely generated projective modules over a commutative local ring are free.Therefore, since for " p and all maximal ideals λ of R w b Z the ring R w,λ is a commutative local ring, we find that T λ pAq is a free R w,λ -module and the same holds for multiples A n .It remains to discuss the local structure at all maximal ideals p of R w bZ p .By Proposition 3.6, the p-adic Tate module T p pAq is a multiple of P w,p .As in all cases considered in Proposition 3.6 the r-th multiple P 'r w,p is a free D w,p -module, the claim follows.Example 3.8.Let w be a finite set of non-real Weil q-numbers, and set P w pxq to be the product of the minimal polynomials P π pxq for π P w.Let 2d be the degree of P w pxq.Since x Þ Ñ q{x permutes the roots of P w pxq, there is a Polynomial Q β pxq P Zrxs, the polynomial with roots β " π `q{π for π P w, with P w pxq " x d Q β px `q{xq.Now Q β pxq has totally real roots of absolute value |β| ă 2 ?q.Let us assume that Q β pxq `1 still is separable with totally real roots, all of which have absolute value bounded by 2 ?q.This certainly can happen when d " 1, i.e. when w " tπu and P π pxq is a quadratic polynomial.Then P w pxq `xd " x d pQ β px `q{xq `1q is still a separable Polynomial with Weil q-numbers as roots.Let w 1 be the set of conjugacy classes of roots of P w pxq `xd , so that P w 1 pxq " P w pxq `xd .It follows that Therefore the the natural map R wYw 1 ÝÑ R w ˆRw 1 is an isomorphism in this case.There are no 'congruences' between π P w and π 1 P w 1 .Let us consider balanced abelian varieties A w in AV w and A w 1 in AV w 1 .Then, for n, n 1 P N, the abelian variety A " A n w ˆAn 1 w 1 is w Y w 1 -locally projective.But T pAq is free as R wYw 1 b Zmodule only if n " n 1 , while T λ pAq is a free R wYw 1 ,λ -module for all n, n 1 .This phenomenon occurs because SpecpR wYw 1 ,λ q is local while SpecpR wYw 1 b Z q is only semi-local and not connected in the example.This illustrates the conceptual advantage of T λ p´q over T p´q, since T λ p´q is local in the sense of Section §1.4.
The above example exploits a case where SpecpR w q is not connected.Now we analyse the connected case.Proposition 3.9.Let w be a finite set of Weil q-numbers.Let A and B be w-locally projective abelian varieties.
(1) If w consists only of ordinary Weil q-numbers, then p o " pF, V, pq is not a maximal ideal of R w and Hom Fq pA, Bq is a projective R w -module.(2) If w contains a non-ordinary Weil q-number, then p o " pF, V, pq is a point in X w " SpecpR w q, the coherent sheaf described by the R w -module Hom Fq pA, Bq is locally free on X w ´tp o u, and there is a Zariski open neighborhood U w of p o such that the rank is constant on U w ´tp o u.
Proof.Since X w " SpecpR w q is the union of the X π " SpecpR π q in Spec ZrF, V s{pF V ´qq, the point p o lies on X w if and only if there is a π P w with p o contained in X π .The latter is equivalent to the existence of a quotient R π F p that sends F, V to 0. In other words, there is a prime ideal of the order R π at which π and q{π lie in the maximal ideal.This happens if and only if π is not ordinary.
The (completed) local structure as an R w -module at a maximal ideal λ above a prime " p (resp. at p above p) is described by Proposition 3.2.Since A and B are w-locally projective, at λ the modules T λ pAq and T λ pBq are projective and To analyse the situatioin at p we may replace A and B by multiples according to Proposition 3.7 and thus assume that A and B are w-locally free.It then follows that T p pAq » D 'n w,p and T p pBq » D 'm w,p for some n, m P N 0 and Hom Fq pA, Bq b Rw R w,p " Hom Dw,p pD 'm w,p , D 'n w,p q (3.3) is isomorphic to a space of matrices with entries in End Dw,p pD w,p q » D op w,p .The first claim follows if D w,p for p " p o is a free R w,p -module.This follows if F R p from (3.1) and if V R p from (3.2).This completes the proof of (1).
We now prove (2).From the proof of (1) we learn that still the coherent sheaf on X w associated to Hom Fq pA, Bq is a locally free sheaf of finite rank on X w ´tp o u.So it is locally away from p o a vector bundle, but the rank a priori may depend on the connected component of X w ´tp o u.Assertion (2) claims that this rank is actually the same on all components of X w ´tp o u whose Zariski closure passes through p o .Therefore it remains to work complete locally at p " p o , i.e. on SpecpR w,po q, where our coherent sheaf is described by (3.3) as isomorphic to the coherent sheaf associated to the R w,po -module pD w,po q 'mn .Note that the R w,po -module structure is insensitive to passing to the opposite ring.
The generic points of SpecpR w,po q correspond to the branches of X w at p o .We must therefore show that D w,po b Q p has the same rank at each generic point of SpecpR w,p 0 q.These generic points correspond to π P w and places v of Qpπq above p with 0 ă vpπq ă vpqq.But after inverting p, the Q p -algebra D w b Q p is an Azumaya algebra of dimension r 2 over Qpwq b Q p , see (2.12), and so the same holds for the base change to R w,p 0 b Q p .It follows that indeed the rank of D w,po b Q p on the generic points of SpecpR w,p 0 q is constant equal to r 2 .Theorem 3.10.Let w be a finite set of Weil q-numbers.We assume that (i) X w " SpecpR w q is connected, and (ii) there exists π P w which is non-ordinary.Let A w be a w-balanced abelian variety.Then the following holds.
(1) If r is even or w only consists of non-real Weil q-numbers, in which case a reduced wbalanced abelian variety s A w exists, then any w-locally projective abelian variety A is isogenous to a power of s A w .
(2) If r is odd and the real conjugacy class of Weil q-numbers t˘?qu is contained in w, then any w-locally projective abelian variety A is isogenous to a power of A w .
Proof.Since w contains a non-ordinary Weil q-number by assumption, the ideal p o " pF, V, pq is a maximal ideal of R w , see Proposition 3.9.By Proposition 3.6 there is an n P N such that The R w -module Hom Fq pA w , Aq is locally free on X w ´tp o u by Proposition 3.9.Moreover, by assertion (2) of Proposition 3.9 its rank is locally constant in a neighborhood of p o .Since X w is connected, such a neighborhood meets every connected component of X w ´tp o u.It follows that Hom Fq pA w , Aq actually corresponds to a vector bundle on X w ´tp o u of some fixed rank.This rank can be computed over R w,po b Q p from the proof of Proposition 3.9 and (3.4) to be `D0 w,po ˘" 2nr 2 .For all maximal ideals λ of R w over a prime " p we deduce that where we used that the rank of the vector bundle Hom Fq pA w , Aq on X w ´tp o u is the same at λ and over a formal punctured neighborhood of p o .In particular, the R w,λ -rank of T λ pAq equals nr and is independent of λ, so that as R w b Z -modules For a moment we assume that r is odd and the real Weil number t˘?qu is contained in w.We set Apπq to be the maximal abelian subvariety of A with Weil support in π, and we let n π denote the multiplicity up to isogeny of the simple B π in Apπq.Let be a prime different from p.By performing base change along R w Ñ Qpwq Ñ Qp ?qq we obtain the rational -adic Tate module V `Ap˘?qq ˘" T pAq b RwbZ Qp ?qq b Q » pQp ?qq b Q q 'nr .
Since r is odd and s ˘?q " 2, it follows that n must be even in this case.Now we come back to the general case.We set (note that we just proved that n{2 is an integer in the second case) if r is odd and w contains the real Weil q-number.
It follows that as R w b Z -modules and thus also as Galois modules Tate's theorem (2.4) and the usual arguments show that A and B are isogenous.
Remark 3.11.The connected components of X w " SpecpR w q correspond to a partition w " w 1 > . . .> w s such that the natural map R w " ÝÑ R w 1 ˆ. . .ˆRws is an isomorphism.Since all connected components X i " SpecpR w i q with a non-ordinary Weil q-number in w i contain the point p o " pF, V, pq, there is at most one such connected component: we call it the non-ordinary component.All the other connected components will be called ordinary connected components.Any w-locally projective abelian variety is the product of w i -locally projective abelian varieties.Theorem 3.10 describes the factor arising from the nonordinary component if that is present.The factor from ordinary components will be explained in Theorem 8.5.

Endomorphism rings of balanced abelian varieties
4.1.Modifying endomorphism rings by local conjugation.In Section §8.1 we will benefit from a certain flexibility to modify the endomorphism ring of a balanced abelian variety.This technique already occured in [Wa69] and we explain it here in a form suitable for our application.
Definition 4.1.Let A and B are abelian varieties in AV Fq .We say that A and B are Tate-locally isomorphic if (i) for all primes " p the Tate modules T pAq and T pBq are isomorphic as Galois modules, (ii) and T p pAq is isomorphic to T p pBq as Dieudonné modules.Versions of this notion have appeared with various terminology, e.g., Zarhin uses 'almost isomorphic' in the context of abelian varieties over finitely generated fields over Q in [Za17].Proposition 4.3.Let w be a finite set of Weil q-numbers.Let A and B be isogenous abelian varieties in AV w .Then, if two of the properties (a) A and B are Tate-locally isomorphic, (b) A is w-locally projective, (c) B is w-locally projective.hold, then also the third holds.The same holds with 'w-locally projective' replaced by 'w-locally free', or (even without assuming A and B being isogenous) 'w-balanced' or 'reduced w-balanced'.
Proof.Projective R w,λ -modules (resp.projective D w,p -modules) are isomorphic to a multiple of a single indecomposable projective module, because R w,λ is a local ring (resp.by Proposition 3.6).Therefore it suffices to compare ranks to deduce (a) from (b) and (c).This is taken care of by the assumption that A and B be isogenous.The other implications are easy.
Lemma 4.4 (compare with [Za17, Lemma 2.3]).Let A and B are abelian varieties in AV Fq .Then A and B are Tate-locally isomorphic if and only if for every prime number (including " p) there exists an isogeny A Ñ B of degree prime to .In particular, if f : A Ñ B and g : B Ñ A are isogenies of coprime degree, then A and B are Tate-locally isomorphic.
Proof.This follows from Tate's theorems recalled in (2.4) and (2.7) as follows.Given an isomorphism ϕ : T pAq Ñ T pBq as Galois or Dieudonné modules, there are isogenies f : A Ñ B and g : B Ñ A such that T pf q and T pgq agree with ϕ respectively ϕ ´1 modulo .It follows that f and g are bijective on -torsion, hence their degree is coprime to .
For the converse direction let f : A Ñ B be an isogeny of degree prime to .Then there is an isogeny g : B Ñ A such that g ˝f is multiplication by an integer prime to , and consequently both T pf q and T pgq are isomorphisms.Definition 4.5.Let w Ď W q be a finite set of Weil q-numbers.An R w -order in a finite dimensional Qpwq-algebra E is an R w -subalgebra O Ď E which is finitely generated as an R w -module and contains a Qpwq-basis of E. Remark 4.6.We will consider R w -orders in E " End Fq pAq b Q for abelian varieties A P AV w .
Since these E are Azumaya algebras over Qpwq, it follows from the Skolem-Noether Theorem [Mi80, IV, Proposition 1.4] that after identifying E » E 1 all isomorphisms to be considered are restrictions of inner automorphisms of E. In this case the term Tate-locally isomorphic coincides with the notion of being Tate-locally conjugate.
Lemma 4.7.Let f : A Ñ B be an isogeny in AV w , and let be a prime number that is coprime to the degree of f .Then the isomorphism maps the R w -order End Fq pAq to an R w -order that agrees with End Fq pBq after completion at , i.e. considered as coherent algebras on SpecpR w q, these orders agree Zariski locally at p q.
Proof.We may complete -adically and use Tate's theorem.We will only discuss the case " p.The case of " p works mutatis mutandis.We are led to consider the isomorphism f p´qf ´1 : End RwbQ pV pAqq " Ý Ñ End RwbQ pV pBqq But by assumption and the proof of Lemma 4.4 the map T pf q : T pAq Ñ T pBq is an isomorphism.The claim follows at once from Tate's theorems (2.4) (resp.(2.7)).
Proposition 4.8.Let A be an abelian variety in AV w .Let O be an R w -order in End Fq pAq b Q that is Tate-locally isomorphic to End Fq pAq.Then there is an abelian variety B that is Tatelocally isomorphic to A and an F q -isogeny f : Proof.Since O and End Fq pAq are both R w -orders of End Fq pAq b Q there is only a finite set Σ of bad prime numbers (including potentially " p) for which End Fq pAq and O do not agree after completion at .We argue by induction on the size #Σ.If Σ is empty, then End Fq pAq " O, and we are done.
We now assume that P Σ and construct an isogeny f : A Ñ B of degree a power of such that A and B are Tate-locally isomorphic and the set of bad primes for B with respect to By Remark 4.6 there is an element φ P End such that f has degree a power of and g has degree prime to .It follows from Lemma 4.4 that B is Tate-locally isomorphic to A. Moreover, Lemma 4.7 applied to f tells us that primes different from at which O and End Fq pAq agree locally, remain primes at which f Of ´1 agrees locally with End Fq pBq.In particular the bad primes for B with respect to f Of ´1 are contained in Σ.
It remains to show that the situation has improved at .For that we compare via g and again by Lemma 4.7, now applicable to since g is of degree coprime to .It follows that This concludes the inductive step and thus proves the proposition.

4.2.
A structure theorem for endomorphism rings of balanced objects.We now describe the endomorphism ring S w " End Fq pA w q for a particular choice of a w-balanced abelian variety A w in AV Fq .A similar description also holds for a reduced w-balanced abelian variety by cancelling a factor of 2 at various places.But first we need to choose some data and fix notation.
Let w be a finite set of Weil q-numbers.
is an Azumaya algebra over Qpwq of degree 2r (so locally a form of M 2r p´q) with the local invariants as specified by Tate's formulas recalled in Theorem 2.2 (2).It follows that the desired S w is an R w -order in S 0 pwq.Waterhouse proves in [Wa69, Theorem 3.13] that there is a simple abelian variety B π P AV π such that End Fq pB π q is a maximal order O π in E π .We set B w " ś π B mπ π and define S pwq :" End Fq pB w q " ź πPw M mπ pO π q Ď ź πPw M mπ pE π q " S 0 pwq.
Then S pwq is a maximal R w -order of S 0 pwq.We next choose a splitting for all " p and for p as Now, for every prime number " p, the proof of the Skolem-Noether theorem (see also Proposition 2.6) shows that there is an isomorphism of Galois modules such that conjugation h p´qh ´1 agrees with And, similarly for p, we have an isomorphism of Dieudonné modules h p : V p pB w q " Ý Ñ pD 0 w q '2 such that conjugation h p p´qh ´1 p agrees with (note that we have passed to the opposite rings) End D 0 w pV p pB w qq op Now we perform the construction of a w-balanced abelian variety from the proof of Theorem 2.7.We choose N coprime to p and large enough such that R w b Zr 1 N p s is a product of Dedekind rings.For all prime numbers N p we may modify h and perforce ψ 0 and ψ so that h induces an isomorphism of integral structures h : pR w b Z q '2r " Ý Ñ T pB w q.
Moreover, we scale suitably h for | N and h p such that these maps restrict to maps h : pR w b Z q '2r ãÑ T pB w q, and h p : T p pB w q ãÑ pD w q '2 .
Using the standard dictionary translating lattices in Tate and Dieudonné modules into isogenies, it follows that there is a corresponding isogeny of degree a product of primes dividing N p, such that h " T phq for all | N and h p " T p phq.It follows by construction and the proof of Lemma 4.4 that A w is indeed w-balanced.Moreover, we have the following description of its endomorphism ring.
Theorem 4.9.With the above notation the endomorphism ring S w " End Fq pA w q of the wbalanced abelian variety A w constructed above sits in the following cartesian square: Proof.The cartesian square is essentially nothing but the obvious cartesian square (fpqc descent) suitably translated by isomorphisms as follows.In view of T pA w q " pR w b Z q '2r for " p (used here only for | N ) and T p pA w q " D '2 w as by construction, Tate's theorem translates the bottom row into the inclusion It remains to identify the right vertical arrow.We may compare A w with B w along h : A w Ñ B w as in the following diagram, in which the top arrow is an isomorphism because degphq is invertible in Zr 1 N p s in combination with Lemma 4.7.
As the bottom square commutes by the definition of h p and h , we have identified the right vertical map in the asserted cartesian square as the map claimed in the theorem.
Remark 4.10.The ring S w constructed in Theorem 4.9 considered as a coherent algebra over SpecpR w q differs from a maximal order in the appropriate Azumaya algebra over Qpwq at most above prime numbers at which R w is singular and those above p.The construction can be performed more carefully so that S w only differs from a maximal order at most in the singularities of SpecpR w q and in the supersingular locus, the vanishing locus of the ideal pF, V, pq.

Representable functors on AV w
In this section we spell out abstract Morita equivalence, originally formulated with abelian categories as for example in [Re75, Chapter 4], in the context of the additive category AV Fq .
Let w Ď W q be a finite set of conjugacy classes of Weil q-numbers, and let A be any object of AV w .Denote by SpAq " End Fq pAq the R w -algebra of F q -endomorphisms of A, and by the contravariant functor represented by A, viewed as valued in the category of left SpAq-modules that are torsion free as Z-modules.Just like AV w , the category Mod Z-tf pSpAqq has a natural R w -linear structure, since the center of SpAq is an R w -algebra in a natural way.Moreover, it is clear that the functor T A p´q is R w -linear.
The main result of this section is Theorem 5.4, a twofold criterion for deciding when (5.1) induces an anti-equivalence.

The maps I
A and G A .The following constructions go back to Waterhouse [Wa69].Let X be any object of AV w , and let H Ď X be any closed subgroup scheme of X. Define the kernel of restriction to H: I A pHq " tf P T A pXq : H Ď kerpf qu " ker `Hom Fq pX, Aq Ñ Hom Fq pH, Aq ˘, a left SpAq-submodule of T A pXq " Hom Fq pX, Aq.Notice that the quotient X{H as group scheme is in fact an abelian variety in AV w and the pull-back via the quotient map ψ H : X Ñ X{H gives an identification of left SpAq-modules as the closed subgroup scheme of X given by the scheme-theoretic intersection of all kernels of elements of M .Notice that the above intersection equals an intersection of finitely many kernels, since M is a finitely generated abelian group.We collect in the next proposition the basic properties of the maps I A and G A .
Proposition 5.1.Let X be any object of AV w .For any closed subgroup schemes H, H 1 of X and any left SpAq-submodules M, M 1 of T A pXq the following properties hold: (1) I A pXrnsq " nT A pXq, for all integers n ě 1, (8) G A pnT A pXqq " Xrns, for all integers n ě 1.
Proof.The properties (1) to (4), ( 7) and ( 8) easily follow from the definition of I A and G A .To see ( 5) and (6), notice that the inclusions are the special cases (4) for M " I A pHq and (3) for H " G A pM q.The corresponding opposite inclusions can be deduced after applying properties (1) and (2) to the inclusions expressed by (3) and (4) respectively.
Remark 5.2.A consequence of Proposition 5.1 is that I A pHq Ď T A pXq has finite index if and only if H Ď X is finite, and that G A pM q Ď X is a finite subgroup if and only if M Ď T A pXq has finite index.
Remark 5.3.We point out that the map G A has already been considered by Waterhouse for X " A, see [Wa69, §3.2].In this circumstance, a finite index submodule M Ď T A pAq " End Fq pAq is an ideal containing an isogeny, and G A pM q, denoted HpM q in loc.cit., is studied via the isogeny A Ñ A{G A pM q.The converse construction I A pHq yields a kernel ideal in the terminology of [Wa69, p. 533].Waterhouse proves that if M is a kernel ideal then the structure of A{HpM q depends only on M as an End Fq pAq-module [Wa69, Theorem 3.11], He moreover proves in [Wa69, Theorem 3.15] that for an abelian variety A whose endomorphism ring is a maximal order in EndpAq b Q any finite index submodule M Ď T A pAq arises as a kernel ideal.

5.2.
A criterion for a functor to be an anti-equivalence of categories.We now state and prove the main theorem of the section.Condition (c) of Theorem 5.4 is inspired by the use of injective cogenerators in embedding theorems for abelian categories, see [Fr64].Note that the category AV w is additive as a full subcategory of the abelian category of all finite type group schemes over the field F q , see [SGA3, Exp.VI, §5.4 Théorème].But AV w is not an abelian category, because an isogeny of abelian varieties is both a monomorphism and an epimorphism in AV w without necessarily being an isomorphism.
Theorem 5.4.Let w Ď W q be a finite subset and let A P AV w be an abelian variety.Let T A : AV w ÝÑ Mod Z-tf pSpAqq, T A pXq " Hom Fq pX, Aq, be the functor represented by A. We consider the following statements.(a) T A is an anti-equivalence of categories: (a 1 ) T A is fully faithful, (a 2 ) T A is essentially surjective.(b) For all X P AV w , the assignments H Þ Ñ I A pHq and M Þ Ñ G A pM q are mutually inverse maps that describe a bijection " H Ď X ; finite subgroup scheme More precisely: (b 1 ) for all X P AV w and all finite subgroup schemes H Ď X, we have H " G A pI A pHqq, (b 2 ) for all X P AV w and all SpAq-submodules M Ď T A pXq of finite index, we have M " I A pG A pM qq.

(c)
A is an injective cogenerator: (c 1 ) for all X P AV w , there exist an n P N and an injective homomorphism X ãÑ A n , (c 2 ) for any X, Y P AV w and any injective homomorphism ϕ : X ãÑ Y , the induced map ϕ ˚: T A pY q T A pXq is surjective.We have the following two chains of equivalences paq ðñ pbq ðñ pcq,
Proof.(a 1 ) ùñ (b 1 ): Let H Ď X be a finite subgroup scheme, and set G " G A pI A pHqq.By Remark 5.2 also G is a finite subgroup scheme.Since the inclusion H Ď G trivially holds, see Proposition 5.1, there is a natural isogeny ϕ : X{H ÝÑ X{G.
By the very definition of G A p´q, it follows that the induced map ϕ ˚: T A pX{Gq ÝÑ T A pX{Hq is an isomorphism.Since by assumption the functor T A p´q is fully faithful, we conclude that ϕ is an isomorphism as well and therefore H " G " G A pI A pHqq.
(b 1 ) ùñ (c 1 ): This is just the special case H " 1.Let tψ 1 , . . ., ψ n u be a set of generators of T A pXq as an SpAq-module and consider the homomorphism ψ : X Ñ A n whose i-th component is ψ i .Since the ψ i 's generate T A pXq, we have kerpψq " G A pT A pXqq.
Assuming (b 1 ), we have 1 " G A pI A p1qq; hence we deduce kerpψq " G A pT A pXqq " G A pI A p1qq " 1, which proves (c 1 ).
(c 1 ) ùñ (a 1 ): We must show that for all abelian varieties X, Y P AV w the map Hom Fq pX, Y q ÝÑ Hom SpAq pT A pY q, T A pXqq (5.3) is bijective.We first show that (5.3) is injective.By assumption pc 1 q there is an injective map ψ : Y ãÑ A n , denote by ψ i its i-th component.If f : X Ñ Y is a homomorphism with 0 " f ˚: T A pY q ÝÑ T A pXq, then ψ i ˝f " 0 for all i, and thus ψ ˝f " 0. But since ψ is injective this implies f " 0.
We next address the surjectivity of (5.3).Let, as above, ψ : Y ãÑ A n be an injection.Since T A pY q is a finitely generated SpAq-module, we may also assume that ψ is chosen so that its components ψ 1 , . . ., ψ n : Y Ñ A generate T A pY q.Construct the quotient pr : A n Ñ Z " A n {Y , which is itself an object of AV w , and choose an injection ι : Z ãÑ A m , which exists again by assumption.Setting σ " ι ˝pr : A n Ñ A m , we obtain a short exact sequence (a co-presentation) We may view σ as an m ˆn matrix σ " ps ij q P M mˆn pSpAqq with entries in SpAq.Let now g : T A pY q Ñ T A pXq be any map of SpAq-modules, and let ϕ : X Ñ A n be the morphism whose i-th component is given by ϕ i " gpψ i q : X Ñ A, where 1 ď i ď n.Let pσ ˝ϕq j be the j-th component of the composition σ ˝ϕ, where 1 ď j ď m.From the SpAq-linearity of g we deduce for any j pσ ˝ϕq j " s jk ψ k q " gppσ ˝ψq j q " 0.
Therefore ϕ factors as ψ ˝f , for a unique map f : X Ñ Y .Since the maps f ˚and g : T A pY q Ñ T A pXq agree on a generating set, hence g " f ˚and (5.3) is indeed surjective.
After having established the equivalence of the assertions (a 1 ), (b 1 ) and (c 1 ) we show that of (a), (b) and (c).We recall that A has Weil support equal to w by assumption.
(a) ùñ (b 2 ): Let X be an object of AV w and let M Ď T A pXq be a submodule of finite index.Thanks to the the fact that T A p´q is essentially surjective by assumption, there is an object Y of AV w and an isomorphism T A pY q » M of SpAq-modules.We deduce an injective SpAq-homomorphism ι : T A pY q ãÑ T A pXq with finite cokernel and which is induced from a homomorphism ϕ : X Ñ Y , because T A p´q is assumed to be fully faithful.
Since ι b Q : Hom Fq pY, Aq b Q Ñ Hom Fq pX, Aq b Q is an isomorphism and A has Weil support equal to w, the map ϕ must be an isogeny.It then follows from the definition that M " I A pkerpϕqq.Applying now property (5) of Proposition 5.1 to the subgroup kerpϕq we find (b) ùñ (c 2 ): Let ϕ : X ãÑ Y be an injective homomorphism in AV w .We have to show that an arbitrary map f : X Ñ A extends to a certain f : Y Ñ A. By composing ϕ with an inclusion Y ãÑ A n , which exists by (c 1 ), we may assume that Y " A n .
Let ϕ i : X Ñ A be the i-th component of ϕ and denote by M the SpAq-submodule of T A pXq generated by all the ϕ 1 , . . ., ϕ n .Poincaré's theorem of complete reducibility implies that X is a direct factor of A n up to isogeny, i.e. there is an abelian subvariety Z Ď A n such that ϕ : X ãÑ A n factors as the inclusion of X ãÑ X ˆZ followed by an isogeny Then f extends as the composition f ˝pr 1 with the first projection to a map X ˆZ Ñ A, and the nonzero multiple g " mf with m " degpψq extends to a morphism g : A n Ñ A. This means precisely that g belongs to M .In this way we see that M has finite index in T A pXq and hence, by assumption (b 2 ), we have Thus there are s i P SpAq with f " ř i s i ϕ i and therefore the map s : A n Ñ A defined as s " where pr i : A n Ñ A is the i-th projection, extends f as desired.
(c) ùñ (a 2 ): Let M be an object of Mod Z-tf pSpAqq and choose a finite presentation By SpAq " T A pAq and additivity of T A p´q the map f comes from a ϕ : A n Ñ A m .Let Y be the image of ϕ and let X " kerpϕq 0 be the reduced connected component of the kernel.Both X and Y are objects of AV w .Let G " kerpϕq{X be the finite group scheme of connected components.Since the functor T A p´q is left exact as a functor on group schemes over F q , we obtain a diagram of SpAq-modules with an exact middle row and exact columns.The map SpAq 'm T A pY q comes from Y ãÑ A m and is surjective by assumption (c 2 ).The same applies to i ˚: SpAq 'n T A pXq which comes from the inclusion i : X ãÑ kerpϕq ãÑ A n .It follows that the map denoted g induces an injective map M ãÑ T A pkerpϕqq.Because T A pGq is finite, we deduce that the composite h h : M ÝÑ T A pXq has finite kernel, and moreover is surjective due to i ˚being surjective.But M is torsion free by assumption, hence h is an isomorphism showing that M lies in the essential image of T A p´q.

The truncated anti-equivalence for finite Weil support
Let w Ď W q be any finite subset fixed throughout the section, and let A w be a w-locally projective abelian variety (see Definition 1.3) with F q -endomorphism ring End Fq pA w q denoted by S w .In this section we show that the functor represented by A w T w : AV w ÝÑ Mod Z-tf pS w q, T w pXq " Hom Fq pX, A w q is an anti-equivalence of categories if A w has full support wpA w q " w.
The ring S w is a finite R w -algebra and free as a Z-module.If λ (resp.p) is a maximal ideal of R w b Z for " p (resp. of R w b Z p ), we introduce notation S w,λ " S w b Rw R w,λ and S w,p " S w b Rw R w,p .
It follows from the local version of Tate's theorem, Proposition 3.2, that T λ p´q and T p p´q induce natural isomorphisms S w,λ " End R w,λ `Tλ pA w q ˘, S w,p " End Rw,p `Tp pA w q ˘op .
6.1.Preliminary lemmata.We now present four lemmata that are needed when proving that T w p´q is an anti-equivalence of categories.The first lemma clarifies an assumption on the support.
Lemma 6.1.Let A be an abelian variety in AV w with support wpAq " w.
(1) T λ pAq is nontrivial for all maximal ideals λ of R w b Z for " p.
(2) T p pAq is nontrivial for all maximal ideals p of R w b Z p .
Proof.Since A has support w, there is an n ě 1 and an isogeny B ˆA1 Ñ A n with B being a w-balanced abelian variety.By Definition 2.8 and Definition 3.1, we have T λ pBq » pR w,λ q '2r .The induced map pR w,λ q '2r » T λ pBq ãÑ T λ pA n q " T λ pAq 'n , is injective, and this shows claim (1).The argument for (2) is similar but uses an isogeny A n Ñ B ˆA1 and the map pD w,p q '2 » T p pBq ãÑ T p pA n q " T p pAq 'n .
We continue by introducing some notation.If

denote by
Mod Z -tf pR w,λ q `resp.Mod Zp-tf pD w,p q the category of finitely generated R w,λ -modules that are free over Z (resp.finitely generated D w,p -modules that are free over Z p ).Similarly, let Mod Z -tf pS w,λ q `resp.Mod Zp-tf pS w,p q be the category of finitely generated left modules over S w,λ that are free over Z (resp.finitely generated S w,p -modules that are free over Z p ).If N is any object of Mod Z -tf pR w,λ q, the formula N ‹ :" Hom R w,λ pN, T λ pA w qq (6.1) defines a contravariant functor on Mod Z -tf pR w,λ q with values in Mod Z -tf pS w,λ q, thanks to the identification S w,λ " End R w,λ pT λ pA w qq that follows from Proposition 3.2 (1).
Lemma 6.2 (Local anti-equivalence at λ).Let w Ď W q be any finite subset, and let A w be a w-locally projective abelian variety with support wpA w q " w.Let λ be a maximal ideal of R w bZ for " p.The functor p´q ‹ : Mod Z -tf pR w,λ q ÝÑ Mod Z -tf pS w,λ q is an anti-equivalence of categories.
Remark 6.3.In (2.3) we recalled that for a prime " p the ring R w b Z is isomorphic to Z rxs{P w pxq.In particular, R w b Z is a Gorenstein ring of dimension one, being a complete intersection.The relevant consequence for us is that any object of Mod Z -tf pR w,λ q is reflexive, that is to say that the dual module functor p´q _ :" Hom R w,λ p´, R w,λ q (6.2) is an anti-equivalence of Mod Z -tf pR w,λ q to itself.For more details see [CS15,Lemma 13].
Proof of Lemma 6.2.For any object N of Mod Z -tf pR w,λ q there is a natural isomorphism which depends functorially on N , where N _ is the dual of N , as defined in (6.2).This is to say that p´q ‹ is isomorphic to the composition of functors Mod Z -tf pR w,λ q p´q _ / / Mod Z -tf pR w,λ q ´bR w,λ T λ pAwq / / Mod Z -tf pS w,λ q.
(6.3) Since T λ pA w q is a free R w,λ -module, and moreover nontrivial by Lemma 6.1 due to the assumption on the Weil support, the functor ´bR w,λ T λ pA w q is a Morita equivalence, and thus p´q ‹ is an anti-equivalence of categories, being a composition of the anti-equivalence (6.2) with an equivalence.
For what concerns the situation at p, for an object N of Mod Zp-tf pD w,p q the formula N ‹ :" Hom Dw,p pT p pA w q, N q defines a covariant functor from Mod Zp-tf pD w,p q to Mod Zp-tf pS w,p q.Notice that S w,p acts on N ‹ from the left thanks to the contravariance of Proposition 3.2 (2) and the natural right action of End Dw,p pT p pA w qq on N ‹ .Lemma 6.4 (Local equivalence at p).Let w Ď W q be any finite subset, and let A w be a w-locally projective abelian variety with support wpA w q " w.Let p be a maximal ideal of R w b Z p .The functor p´q ‹ : Mod Zp-tf pD w,p q ÝÑ Mod Zp-tf pS w,p q, is an equivalence of categories.
Proof.Since A w is assumed w-locally projective, by Proposition 3.7 there is an integer n ě 1 such that A n w is w-locally free.The functors p´q ‹ for A w and the power A n w are linked by a Morita equivalence.Therefore we may assume withlout loss of generality that T p pA w q is a free D w,p -module.
Let T p pA w q be free of rank n over D w,p .By Lemma 6.1, due to the assumption on the Weil support, we have n ě 1.A choice of a basis determines an isomorphism T p pA w q » D 'n w,p and a natural isomorphism N ‹ " Hom Dw,p pT p pA w q, N q » Hom Dw,p pD 'n w,p , N q " N 'n with the S w,p action on N ‹ corresponding to the M n pD w,p q-action on N 'n under the isomorphism S w,p " End Dw,p pT p pA w qq op » End Dw,p pD 'n w,p q op " M n pD w,p q of Proposition 3.2 (2) and induced by that very same choice of basis.Hence the functor p´q ‹ is isomorphic to a Morita equivalence restricted to modules that are free and finitely generated as Z p -modules.While Lemmata 6.2 and 6.4 above enter in the proof of fully faithfulness of T w p´q, the next lemma is needed to show that T w p´q is essentially surjective.Lemma 6.5.Let A ãÑ B be an injective homomorphism of abelian varieties in AV Fq , and let " p be a prime number.Then the following holds.
(1) The natural map T pAq Ñ T pBq is injective and cotorsion free, i.e. the cokernel is a free Z -module.(2) The natural map T p pBq Ñ T p pAq is surjective.
Proof.The quotient C " B{A exists in the category of finite type group schemes over SpecpF q q and is an abelian variety.For all n P N multiplication by n is an isogeny, hence surjective.It follows from the snake lemma in the abelian category of finite type group schemes over SpecpF q q, see [SGA3, Exp.VI, §5.4 Théorème], that we have an exact sequence 0 ÝÑ Arns ÝÑ Brns ÝÑ Crns ÝÑ 0 (6.4) of finite flat group schemes.By passing to the limit for n " m one deduces from (6.4) the exact sequence 0 ÝÑ T pAq ÝÑ T pBq ÝÑ T pCq ÝÑ 0.
Now (1) follows because the cokernel T pCq is a free Z -module.
(2) Since T p pAq is the limit of the Dieudonné modules for the system of Arp m s and the Dieudonné module functor is exact on finite flat group schemes we also have an exact sequence 0 ÝÑ T p pCq ÝÑ T p pBq ÝÑ T p pAq ÝÑ 0. 6.2.Existence and characterisation of injective cogenerators.We are now ready to prove the main result of the section.Theorem 6.6.Let w Ď W q be a finite subset, and let A w in AV w be an abelian variety.Then the following are equivalent.(a) A w is w-locally projective with support wpA w q " w.

(b)
A w is an injective cogenerator for AV w .(c) The functor1 T w : AV w ÝÑ Mod Z-tf pS w q, T w pXq " Hom Fq pX, A w q is an anti-equivalence of categories.
Proof.We start by showing (a) ùñ (c).So A w is w-locally projective with support w.The first thing to show is that T w p´q is fully faithful.To this purpose let X, Y be abelian varieties in AV w , and consider the map τ : Hom Fq pX, Y q ÝÑ Hom Sw pT w pY q, T w pXqq induced by T w p´q.Both source and target of τ are finitely generated R w -modules.Hence it suffices to verify that τ b R w,λ and also τ b R w,p are bijective for all maximal ideals λ of R w b Z , for " p, and all maximal ideals p of R w b Z p respectively.The case λ: consider the commutative diagram

HompTate,Tateq
Hom R w,λ `Tλ pXq, T λ pY q ˘/ / Hom S w,λ `Tλ pY q ‹ , T λ pXq ‹ whose arrows all are the natural ones.Thanks to the λ-adic version of Tate's isomorphism as in Proposition 3.2, and to flat localization along R w Ñ R w,λ , all vertical arrows of the diagram are isomorphisms.Thanks to Lemma 6.2, the bottom horizontal arrow is an isomorphism, hence the same is true for τ b R w,λ .
The case p: Similarly, the vertical and the bottom arrows of the commutative diagram Hom Fq pX, Y q b Rw R w,p τ bRw,p / / Tate Hom Sw `Tw pY q, T w pXq ˘bRw R w,p Hom Sw,p `Tw pY q b Rw R w,p , T w pXq b Rw R w,p HompTate,Tateq Hom Dw,p `Tp pY q, T p pXq ˘/ / Hom Sw,p `Tp pY q ‹ , T p pXq ‹ ȃre isomorphisms, thanks to the p-adic version of Tate's isomorphism as in Proposition 3.2, flat localization along R w Ñ R w,p , and Lemma 6.4.This implies that τ b R w,p is an isomorphism, completing the proof that T w p´q is fully faithful.
We now show that T w p´q is essentially surjective.Since we already know that T w p´q is fully faithful, it is enough to check that T w p´q satisfies condition (c 2 ) of Theorem 5.4, i.e. we need to show that for all injections i : X ãÑ Y in AV w the induced map i ˚: T w pY q ÝÑ T w pXq (6.5) is surjective.We accomplish this by showing that i ˚b Z is surjective for all primes .
The case " p: by Tate's isomorphism (2.4) the scalar extension i ˚b Z is identified with the induced map Hom RwbZ pT pY q, T pA w qq ÝÑ Hom RwbZ pT pXq, T pA w qq, ϕ Þ Ñ ϕ ˝T piq on -adic Tate modules.Define the R w b Z -module M by exactness of the sequence 0 ÝÑ T pXq ÝÑ T pY q ÝÑ M ÝÑ 0.
Since i is injective, Lemma 6.5 (1) shows that M is free as Z -module and hence reflexive as R w b Z -module, see Remark 6.3.The Ext-sequence Hom RwbZ pT pY q, T pA w qq ÝÑ Hom RwbZ pT pXq, T pA w qq ÝÑ Ext 1 RwbZ pM, T pA w qq shows that surjectivity of i ˚b Z follows from the vanishing of Ext 1 RwbZ pM, T pA w qq.Since R w b Z is the product of the local rings R w,λ , we have

Ext 1
RwbZ pM, T pA w qq " Since A w is locally projective, Proposition 3.7 shows that T λ pA w q is a free R w,λ -module.Now the vanishing Ext 1 R w,λ pM b R w,λ , T λ pA w qq " 0 is a consequence of [CS15, Lemma 17] and the fact that R w,λ is a Gorenstein ring of dimension 1 as the localization of R w b Z .
The case " p: using Tate's isomorphism (2.7), the surjectivity of i ˚b Z p translates into the surjectivity of Hom Dw pT p pA w q, T p pY qq ÝÑ Hom Dw pT p pA w q, T p pXqq, which is an immediate consequence of Lemma 6.5 (2) and the fact that T p pA w q is a projective D w -module.The equivalence of (c) and (b) is proven as part of Theorem 5.4.It remains to show that (b) implies (a).For that we pick an auxiliary abelian variety A 1 w that is w-balanced.Such an A 1 w exists by Theorem 2.7, and A 1 w is w-locally projective with support w.By what we have already shown, A 1 w is an injective cogenerator.Therefore there exist an n and an embedding i : A w ãÑ pA 1 w q n according to Theorem 5.4 (c 1 ).Now we apply Theorem 5.4 (c 2 ) to this embedding, but with respect to the injective cogenerator A w .It follows that the map Hom Fq ppA 1 w q n , A w q Hom Fq pA w , A w q, ϕ Þ Ñ ϕ ˝i is surjective.A preimage of the identity is a retraction pA 1 w q n Ñ A w which shows that A w is a direct factor of pA 1 w q n .In particular, T λ pA w q (resp.T p pA w q) is a direct factor of a projective R w,λ (resp.a projective D w,p )-module and hence is itself projective.This shows that A w is w-locally projective.
For all π P w, applying Theorem 5.4 (c 1 ) to a simple abelian variety B π in the isogeny class associated to π yields an embedding B π ãÑ A n w for some n.In particular, π is contained in the support of A w .This completes the proof.Remark 6.7.In an appendix to [La01] Serre proves a version of Theorem 6.6 for w " tπu and an ordinary Weil number π associated to an ordinary elliptic curve E such that R π is a maximal order in Qpπq.In contrast to Deligne's proof in [De69], Serre's proof uses the covariant functor Hom Fq pE, ´q, so no multiplicities are used.
We reprove in Theorem 8.4 Deligne's result for the category of all ordinary abelian varieties.Also in this case the representing object A ord w is isogenous to ś πPw B π , so again no multiplicities are necessary.However, the general case requires multiplicities.In Section 9 we dicuss completely the necessity of multiplicities in the case of AV π for a Weil number π associated to an elliptic curve.The crucial result concerning multiplicities is Theorem 3.10.Corollary 6.8.Let A in AV w be an injective cogenerator.Then A is a direct factor of a power of a w-balanced abelian variety.
Proof.This was proved at the end of the proof of Theorem 6.6.Proposition 6.9.Let w Ď W q be a finite subset, and let A w in AV w be a w-balanced abelian variety.Then, for any X in AV w , the Z-rank of T w pXq " Hom Fq pX, A w q equals rk Z T w pXq " 4r dimpXq.
Proof.Since both sides of the equation are isogeny invariant and additive with respect to products of abelian varieties, it suffices to consider F q -simple objects X " B π for π P w.Then rk Z pT w pB π qq " m π rk Z `End Fq pB π q ˘" 2r s π ¨s2 π rQpπq : Qs " 4r dimpB π q, where the last equality uses Theorem 2.2 (1) from Honda-Tate theory.
6.3.Realizing R w as the center.If w " tπu consists of a single ordinary Weil q-number π, then Waterhouse shows in [Wa69, Chapter 7] that the minimal endomorphism ring R π arises as the endomorphism ring of a simple, ordinary abelian variety B π .We will reprove this later as part of Proposition 8.3.In this section we will show that R w agrees with the center of End Fq pAq for any injective cogenerator A in AV w .This verifies the claim made in the introduction that the center R q of S q can be described explicitly in terms of Weil q-numbers.
Proposition 6.10.Let A P AV w be an injective cogenerator.Then the natural map R w ÝÑ End Fq pAq identifies R w with the center of End Fq pAq.
Proof.We first prove the result for a w-balanced abelian variety A w .In order to show that the map R w Ñ ZpEnd Fq pA w qq is an isomorphism, where Zp´q denotes the center, it suffices to show this after completion at all prime numbers.We deduced from Tate's theorems (2.4) and (2.7) that for all primes ‰ p End Fq pA w q b Z " End RwbZ pT pA w qq » End RwbZ `pR w b Z q '2r ˘" M 2r pR w b Z q, End Fq pA w q b Z p " End Dw pT p pA w qq op » End Dw pD '2 w q op " M 2 pD w q.
In both cases we determine the center as indeed R w b Z (resp.R w b Z p , see Lemma 2.1).Now let A be an arbitrary injective cogenerator.Let S " End Fq pAq and denote the antiequivalence Hom Fq p´, Aq by T p´q.Then the composition of the natural injective maps R w ãÑ ZpSq ãÑ ZpEnd S pT pA w qqq " ZpEnd Fq pA w q op q is an isomorphism due to the discussion of the w-balanced case.The result follows.
Remark 6.11.The converse to Proposition 6.10 does not hold for q " p 4 and π " p 2 .The simple object in AV π is an elliptic curve B π with EndpB π q a maximal order in the quaternion algebra over Q ramified in p8.Then R π " Z equals the center of EndpB π q.But Homp´, B π q is not an anti-equivalence according to [JKP+18, Theorem 1.1].Since π is non-ordinary, Theorem 3.10 says that all injective cogenerators of AV π are isogenous to a power of a reduced π-balanced object.Thus injective cogenerators of AV π have even multiplicity because of s m π " r{s π " 2.
7. Compatible truncated anti-equivalences 7.1.Maximal subgroup with partial Weil support.Recall that for an abelian variety A over F q we set SpAq " End Fq pAq, and that SpAq is an R w -algebra if A P AV w .
Proposition 7.1.Let v Ď w Ď W q be finite sets of Weil numbers, and let A w P AV w be an injective cogenerator.Then the subgroup generated by all images A v,w :" ximpf q ; f : X Ñ A w , X P AV v y Ď A w satisfies the following: (1) A v,w belongs to AV v and is an abelian subvariety of A w .
(2) A v,w is an injective cogenerator in AV v .
(3) Restriction to A v,w induces a natural surjection pr v,w : SpA w q SpA v,w q that is an R w R v algebra map.(4) We set T w pXq " Hom Fq pX, A w q and T v,w pXq " Hom Fq pX, A v,w q.The following diagram naturally commutes: Here the right inclusion is defined by pulling back the SpA v,w q-module structure via pr v,w to a SpA w q-module structure.(5) If A w is w-balanced, then A v,w is v-balanced.
(2) Being an injective cogenerator means that property property (c) of Theorem 5.4 holds, so this holds for A w .We are going to show that property (c) also holds for A v,w .Now (c 1 ) holds because for an arbitrary X P AV v there is an injection X ãÑ A n w for a suitable n that factors through A n v,w Ď X n w by the definition of A v,w .To show (c 2 ) we start with an injection X ãÑ Y in AV w and a homomorphism f 0 : X Ñ A v,w .By composing with i : A v,w ãÑ A w we can extend f " i ˝f0 to a g : Y Ñ A w since A w is an injective object, i.e.A w satisfies property (c 2 ).By definition g factors through g 0 : Y Ñ A v,w and g 0 extends f 0 because i is injective.
(3) Restriction with i : A v,w ãÑ A w yields a surjection by Theorem 5.4 (c 2 ) pr v,w : SpA w q " Hom Fq pA w , A w q Hom Fq pA v,w , A w q " Hom Fq pA v,w , A v,w q " SpA v,w q with the next to last equality due to the definition of A v,w .Here pr v,w pgq for a homomorphism g : A w Ñ A w is the unique homomorphism so that the following commutes: The map pr v,w is indeed a ring homomorphism as for a composition f ˝g of f, g P SpA w q the composition pr v,w pf q ˝pr v,w pgq completes the square as in the following diagram: Moreover, the map pr v,w is an R w R v algebra map, because Frobenius and Verschiebung are natural with resepct to i : A v,w ãÑ A w .
Assertion (4) follows from the natural equality for every X P AV v Hom Fq pX, A v,w q " Hom Fq pX, A w q, since every morphism f : X Ñ A w takes values in the subvariety A v,w Ď A w .Now we show (5).If A w is w-balanced, then all simple abelian varieties B π for π P w occur in A w with multiplicity m π .Since up to isogeny A v,w consists of all isogeny factors B π of A w with π P v, we deduce that A v,w has the same multiplicities m π for all π P v. Thus A v,w is isogenous to a v-balanced abelian variety A v .Being an injective cogenerator, A v,w is v-locally projective by Theorem 6.6.Since A v is also v-locally projective, now Proposition 4.3 shows that A v,w and A v are Tate-locally isomorphic.Again Proposition 4.3 then shows that A v,w is v-balanced, since A v is v-balanced.
7.2.The direct system.In order to prove Theorem 1.1 we construct a direct system A " lim Ý Ñ w A w consisting of abelian varieties A w indexed by finite subsets w Ď W q of Weil q-numbers such that for all w the functors T w : AV w ÝÑ Mod Z-tf pS w q, T w pXq " Hom Fq pX, A w q are anti-equivalences, that moreover are naturally compatible among each other.
Proof of Theorem 1.1.For any finite subset w Ď W q let Zpwq be the set of isomorphism classes rAs of w-balanced abelian varieties The set Zpwq is not empty by Theorem 2.7.The elements of Zpwq all belong to the same isogeny class, and so Zpwq is finite, since there are only finitely many isomorphism classes of abelian varieties over a finite field lying in a given isogeny class (in fact, Zarhin shows in [Za77, Theorem 4.1] that finiteness holds for isomorphism classes of abelian varieties of fixed dimension).
For any pair v Ď w of finite subsets of Weil q-numbers, we construct a map indexed by finite subsets w Ď W q .Since the sets Zpwq are finite and non-empty, a standard compactness argument shows that the inverse limit is not empty: We choose a compatible system z " pz w q P Z of isomorphism classes of abelian varieties.Now we would like to choose abelian varieties A w in each class z w and inclusions for every v Ď w that are isomorphic to the inclusions A v,,w Ď A w discussed in Proposition 7.1 in a compatible way: for u Ď v Ď w we want Because the set of Weil numbers is countable, we may choose a cofinal totally ordered subsystem of finite subsets of W q w 1 Ď w 2 Ď . . .Ď w i Ď . . ., where cofinal implies Ť i w i " W q .Working with this totally ordered subsystem, we can construct a direct system A 0 " pA w i , ϕ w j ,w i q of abelian varieties as desired by induction.If A w i is already constructed, then we choose A w i`1 in z w i`1 and deduce from ζ w i ,w i`1 pz w i`1 q " z w i that there is an injective homomorphism ϕ w i`1 ,w i : A w i Ñ A w i`1 as desired.By construction, the map ϕ w i`1 ,w i is an isomorphism of A w i with the abelian subvariety A w i ,w i`1 of A w i`1 as discussed in Proposition 7.1.
Once A 0 is constructed, we may identify all transfer maps of the restricted system A 0 with inclusions.Let now w be a general finite subset of W q .Since Ť i w i " W q is a union of a totally ordered system of w i , we find w Ď w i for all large enough i.Then we can define the abelian variety A w by means of the construction of Proposition 7.1 as the abelian subvariety This choice is well defined, i.e. independent of i " 0. Furthermore, there are compatible transfer maps ϕ v,w : A v Ñ A w for all v Ď w that lead to the desired direct system A " pA w , ϕ w,v q.
In the sense of ind-objects we have A 0 » A and so A 0 would suffice for Theorem 1.1.But we prefer the more aesthetic ind-system A indexed by all finite subsets w Ď W q .
Let X be any element of AV Fq , and set T pXq " Hom Fq pX, Aq " lim Ý Ñ w Hom Fq pX, A w q " lim Ý Ñ w T w pXq.
The groups Hom Fq pX, A w q are stable when w is large enough.More precisely, if w, w 1 are finite subsets of Weil q-numbers with wpXq Ď w Ď w 1 , then the map ϕ w 1 ,w ˝´: Hom Fq pX, A w q ÝÑ Hom Fq pX, A w 1 q is an isomorphism, cf.Proposition 7.1 (4).Moreover, T p´q restricted to AV w recovers the functor T w p´q of Theorem 6.6 constructed using the object A w of A. Furthermore, the functor T w p´q induces an anti-equivalence between AV w and Mod Z-tf pS w q by Theorem 6.6 because A w is w-balanced.
We next show that T p´q is an R q -linear functor.We denote the linear Frobenius isogeny for an abelian variety X over F q by π X : X Ñ X. Observe that, since the Frobenius isogeny is a natural transformation, for any finite subset w Ď W q and for any f P Hom Fq pX, A w q the diagram is commutative.This implies that, for w sufficiently large so that T pXq " Hom Fq pX, A w q, w large, the action F pf q " π Aw ˝f on f P T pXq of the image of F in R w Ď S w is given by the morphism induced by the Frobenius isogeny π X via functoriality of T T pπ X q : f Þ Ñ f ˝πX " π Aw ˝f.
A similar consideration with the isogeny Verschiebung shows that indeed T p´q is R q -linear.Compatibility in w shows that T p´q induces an anti-equivalence Complemented by the rank computation of Proposition 6.9, this is precisely the claim of Theorem 1.1 and so its proof is complete.
Remark 7.2.When reduced w-balanced abelian varieties exist, i.e. if we avoid π " ˘?q when r is odd, then the above proof of Theorem 1.1 can be applied mutatis mutandis to construct an anti-equivalence of categories represented by an ind-abelian variety s A " p s A w , ϕ w,w 1 q consisting of reduced w-balanced abelian varieties.This anti-equivalence yields modules Hom Fq pX, A w q of Z-rank 2r dimpXq.For r " 1, this reproves the main result of [CS15].7.3.Weil restriction of scalars and base change.Another form of compatibility can be established with respect to Weil restriction of scalars and with respect to base change.We will not need these compatibilities here and therefore only report the results, all of which have quite formal proofs.
Proposition 7.3.Let w be a finite set of Weil q m -numbers, and set m ?w :" tπ ; π m P wu.(1) Let A P AV w be an injective cogenerator.Then the Weil restriction RA :" R F q m |Fq pAq is an injective cogenerator for AV m ?w , and for all X P AV m ?w we have a natural isomorphism of functors in X T A pX b Fq F q m q » T RA pXq.(2) Let A P AV m ?w be an injective cogenerator.The base change A m :" A b Fq F q m is an injective cogenerator for AV w , and for all X P AV w we have a natural isomorphism of functors in X T A pR F q m |Fq pXqq " T Am pXq.
(3) Let A m ?w P AV m ?w be m ?w-balanced.Then the scalar extension One can further show that an ind-representing object A " pA ω q ωĎWq provides a similar indrepresenting object A b Fq F q m after base changing those with index m ?w for subsets w Ď W q m .

The commutative case and optimal rank
We discuss when a modified anti-equivalence takes values in modules over a commutative ring.
8.1.Recovering Deligne's result: the ordinary case.In this section we will be concerned with lowering multiplicities of injective cogenerators for sets of ordinary Weil q-numbers.
Lemma 8.1.For a finite set w of ordinary Weil q-numbers, we have D w » M r pR w b Z p q.
Proof.It suffices to determine the structure locally at maximal ideals p of R w b p, because R w b Z p is the product of these localizations.As w consists only of ordinary Weil q-numbers either F R p of V R p.The claim D w,p " M r pR w,p q now follows from the local structure results (3.1) and (3.2) obtained in the proof of Proposition 3.6.
Proposition 8.2.Let w be a finite set of ordinary Weil q-numbers.Then there exist a wbalanced abelian variety A w and an R w -linear isomorphism End Fq pA w q » M 2r pR w q.
Proof.Let A 1 w be an arbitrary w-balanced abelian variety.Then End Fq pA 1 w q b Q is an Azumaya algebra of degree 2r over its center Qpwq.This center is a product of number fields.The localglobal principle for the Brauer group, see [BHN32], shows that End Fq pA 1 w q b Q is split as an Azumaya algebra over Qpwq, because all local invariants are trivial by Tate's formula as recalled in Theorem 2.2 (2).Here we use that w only contains ordinary Weil numbers.The spliting translates into a Qpwq-algebra isomorphism End Fq pA 1 w q b Q » M 2r pQpwqq.Moreover, the integral local structure of S 1 w " End Fq pA 1 w q can be deduced from Tate's theorems (2.4) and (2.7) and Lemma 8.1 as S 1 w b Z " End RwbZ pT pA 1 w qq » End RwbZ `pR w b Z q '2r ˘" M 2r pR w b Z q, S 1 w b Z p " End Dw pT p pA 1 w qq op » End Dw pD '2 w q op " M 2 pD w q " M 2r pR w b Z p q.
Proposition 4.8 applied to the R w -order O " M 2r pR w q shows that there exists an abelian variety A w that is Tate-locally isomorphic to A 1 w and such that End Fq pA w q » M 2r pR w q.The abelian variety A w is also w-balanced by Proposition 4.3.Proposition 8.3.Let w be a finite set of ordinary Weil q-numbers.Then there exists an abelian variety A ord w with the following properties.(i) The Weil support of A ord w is equal to w, (ii) the natural inclusion R w Ă End Fq pA ord w q is an isomorphism, (iii) A ord w is w-locally projective, and (iv) 2 dimpA ord w q " rQpwq : Qs.In particular, the corresponding contravariant functor T A ord w : AV w ÝÑ Mod Z-tf pR w q, T A ord w pXq " Hom Fq pX, A ord w q gives an R w -linear anti-equivalence of categories.
Proof.We choose a w-balanced A w and fix an isomorphism EndpA w q » M 2r pR w q as in Proposition 8.2.Let e i P EndpA w q be the idemponent that corresponds in EndpA w q » M 2r pR w q to the matrix with a single 1 at the ith diagonal entry.These idempotents e 1 , . . ., e 2r commute and sum to the identity.Let A i be the image of e i : A w Ñ A w .It follows formally as in any additive category that A w » A 1 ˆ. . .ˆA2r .
Moreover, let e ij P EndpA w q » M 2r pR w q correspond to the elementary matrix with a single 1 in the ith row and jth column.Then e ij | A j : A j Ñ A i is an isomorphism, so all direct factors of A w are mutually isomorphic.We define A ord w " e 1 pA w q " A 1 , so that A w » pA ord w q 2r .In particular, the support of A ord w is the support of A w and equal to w.Moreover, we can compute as R w -algebras EndpA ord w q " e 1 EndpA w qe 1 " R w .Now A ord w is w-locally projective because a multiple A w " pA ord w q 2r is w-locally projective, and direct summands of projective modules are again projective.As A ord w has full supprt, by Theorem 6.6 we have that A ord w indeed represents the desired anti-equivalence of the claim.In order to compute the dimension of A ord w , we note that all π P w are ordinary and thus have commutative E π " End Fq pB π q.Hence s π " 1, and the multiplicity of B π in A w is m π " 2r.It follows that A ord w is isogenous to ś πPw B π .The dimension formula is a consequence of Theorem 2.2 (1) and Qpwq " ś πPw Qpπq.
We can now reprove Deligne's main result of [De69].We choose to state a contravariant form in accordance with our overall choice.By precomposing with the functor dual abelian variety we can pass from contravariant to covariant equivalences.
Recall that AV ord Fq denotes the full subcategory of AV Fq consisting of ordinary abelian varieties.Similarly, we denote by R ord q the projective system pR w , r v,w q restricted to finite sets w of ordinary Weil q-numbers.
Theorem 8.4 (Deligne [De69]).Let q " p r be a power of a prime number p.There exists an indabelian variety A ord " pA ord w , ϕ w,w 1 q indexed by finite sets w Ď W ord q of ordinary Weil q-numbers, such that A ord w is w-locally projective, the transition maps are inclusions, and End Fq pA ord w q " R w .The contravariant and R ord q -linear functor T ord : AV ord Fq ÝÑ Mod Z-tf pR ord q q, T ord pXq " Hom Fq pX, A ord q is an anti-equivalence of categories which preserves the support.Moreover, the Z-rank of T ord pXq is equal to 2 dimpXq.
Proof.The construction of Proposition 7.1 applied to an A ord w produced by Proposition 8.3 yields an injective cogenerator A ord v,w such that End Fq pA ord v,w q is an R v -algebra that is a quotient of R w " End Fq pA ord w q.It follows that also End Fq pA ord v,w q " R v .Now the construction of the ind-abelian variety A ord works as in the proof of Theorem 1.1 by replacing w-balanced abelian varieties by varieties A ord w that are w-locally projective with End Fq pA ord w q " R w .This proves the claim.
The reduction of multiplicity achieved in Proposition 8.3 allows to complete the structure theory of w-locally projective abelian varieties begun in Section §3.Recall that there is a tensor product construction between abelian varieties with multiplication by a ring R and certain Rmodules, see Serre's appendix to [La01] and [JKP+18, §4.1].
Theorem 8.5.Let w be a finite set of ordinary Weil q-numbers, and let A ord w be a w-locally projective abelian variety with R w " End Fq pA ord w q as in Proposition 8.3.Then any w-locally projective abelian variety A is of the form A » A ord w b Rw P where P " Hom Fq pA ord w , Aq is a finitely generated projective R w -module.
Proof.Proposition 3.9 (1) shows that Hom Fq pA ord w , Aq is indeed projective.The general properties of the tensor product yield an evaluation map A ord w b Rw Hom Fq pA ord w , Aq ÝÑ A. In order to show that this map is an isomorphism, it sufffices to show this locally on -adic and p-adic Tate modules.There it follows because A and A ord w are w-locally projective and T pA ord w q (resp.T p pA ord w q) is locally free of rank 1 (resp.the unique indecomposable projective module, see Proposition 3.6).
Remark 8.6.Using Theorem 3.10 and Theorem 8.5 one obtains a complete description of the isogeny classes of AV w containing an injective cogenerator for AV w .See also Remark 3.11.8.2.Injective cogenerators with commutative endomorphism ring.Let W Ď W q be a (possibly infinite2 ) set of Weil numbers.The proof of Theorem 1.1 adapts immediately to provide an ind-abelian variety A W that ind-represents an anti-equivalence of AV W with Mod Z-tf pS W q, where S W equals the pro-ring End Fq pA W q.
We would like to determine all sets W for which there is such an A W as above with S W commutative.
Recall that we denote by r the degree of F q over F p .
Proposition 8.7.Let π be a Weil q-number, and let A π be an injective cogenerator for AV π that has a commutative ring of endomorphisms S π .Then (1) π is ordinary, or (2) r " 1 and π is not the real conjugacy class of Weil p-numbers t˘?pu.
Proof.In order to have a commutative ring of endomorphisms, A π must be F q -simple and s π " 1.
The last condition also imples that Qpπq has no real places and thus that a reduced π-balanced object exists.If π is not ordinary, then by Theorem 3.10 (note that SpecpR π q is connected), the abelian variety A π must be isogenous to a power of a reduced π-balanced abelian variety.The variety A π being F q -simple, this in particular implies s m π " 1, and thus r " s π ¨s m π " 1.
We conclude that the examples of commutative cases as presented in Section §1.5 of the introduction cover all commutative cases.Proposition 8.8.The only sets of Weil q-numbers W with a commutative S W and an indrepresentable R W -linear anti-equivalence AV W Ñ Mod Z-tf pS W q are contained in (i) the set W ord Ď W q of ordinary Weil q-numbers, or (ii) the set W com p " W p zt˘?pu of non-real Weil p-numbers.
Proof.Let A W be an inductive system of abelian varieties from AV W that ind-represents an antiequivalence of categories as in the proposition.By a diagonal process based on the technique of Proposition 7.1, we may assume that A W " pA w q wĎW is indexed by finite subsets of Weil q-numbers in W , and such that A w represents the restriction of the anti-equivalence to AV w .It follows from Theorem 6.6 that, for all π P W , the abelian variety A π must be an injective cogenerator for AV π with a commutative ring of endomorphisms.Now the claim follows from Proposition 8.7.
8.3.Lattices of optimal rank.A noticeable feature of the main result Theorem 1.1 is the necessity to work with lattices whose rank is a multiple of the first Betti number.
Proposition 8.9.Let Λ : AV Fq Ñ Mod Z-tf pZq be an additive, contravariant functor that attaches to any abelian variety over F q a free Z-module of finite rank.Assume that there is a constant γ ą 0 such that rk Z pΛpXqq " γ ¨dimpXq for any X in AV Fq .Then γ is divisible by 2 lcmpr, 2q.
Proof.Let π be a Weil q-number, and B π a simple object of AV Fq associated to π.The Q-vector space ΛpB π q b Q defines, via functoriality of Λ, a right representation of E π " End Fq pB π q b Q.
Since E π is a division algebra, E π is the unique simple object in the category of right representations of E π .It follows that ΛpB π q b Q is isomorphic to a multiple of E π .Therefore γ ¨dimpB π q " dim Q ΛpB π q b Q is a multiple of (using Theorem 2.2 (1)) dim Q E π " s 2 π rQpπq : Qs " 2s π dimpB π q.Hence γ is divisible by the least common multiple of all 2s π for all Weil q-numbers π.It was discussed in Section §2.4 that s π always divides lcmpr, 2q.
To complete the proof it is enough to show that ‚ for any r ą 2 there exists a Weil q-number π 1 such that s π 1 " r, and ‚ for any r ě 1 there exists a Weil q-number π 2 with s π 2 " 2. The first statement is the content of Lemma 8.10 below.The second one is proved after checking that any root π 2 of x 2 ´q satisfies the required properties.
Lemma 8.10.Let r ą 2, and let π be a root of the polynomial f pxq " x 2 ´px `q, which is irreducible in Qrxs.Then π defines a Weil q-number such that the index s π of E π is r.
Proof.The discriminant of f pxq is p 2 ´4q ă 0. Thus f pxq is irreducible with two complex conjugate roots and hence π is a Weil q-number.
In order to compute s π we must compute the order of the local invariants at p-adic places as in Theorem 2.2 (2).The assumption r ą 2 ensures that the Newton polygon of x 2 ´px `q with respect to the p-adic valuation v p has two different (negative) slopes 1 and r ´1.Therefore f pxq splits in Q p rxs into a product of two distinct linear factors corresponding to two distinct primes of Qpπq above p.Moreover, by comparing with the slopes, we can choose a prime p | p such that v p pπq " 1.By Theorem 2.2 (2), the local invariant of the division ring E π at p is 1{r pmod 1q, which suffices to ensure s π " r.

Abelian varieties isogenous to a power of an elliptic curve
Let E be an elliptic curve over F q and π " π E the associated Weil q-number.The category AV π defined in Section §1.2 is the full subcategory of AV q whose objects are the abelian varieties over F q isogenous to a power of E. This category is the object of study of [JKP+18].In this stimulating paper the authors give a characterization of those elliptic curves E for which the functor T E : AV π ÝÑ Mod Z-tf pEnd Fq pEqq, T E pXq " Hom Fq pX, Eq is an anti-equivalence of categories.Their main result [JKP+18, Theorem 1.1] says that T E is an anti-equivalence precisely in the following cases (recall r " rF q : F p s): ‚ π is ordinary and EndpEq " R π ; ‚ π is supersingular, r " 1, and EndpEq " R π , or ‚ π is supersingular, r " 2, and EndpEq is of Z-rank 4.
Remark 9.1.This result provides the complete list of Weil numbers π for which an injective cogenerator of dimension 1 for AV π exists.
For those supersingular Weil numbers π associated to an elliptic curve that are left out from the treatment of [JKP+18], it is natural to ask what is the minimal dimension of an injective cogenerator of AV π .To this purpose we draw the following consequence of Theorem 3.10.and for a choice of a reduced balanced Āπ in order to show what the obstacle is and how Āπ circumvents the local problem at p. Let K " Q p pπq be the completion of Qpπq at the unique prime above p, fix an embedding W pF q q ãÑ K and denote by o K the image of W pF q q.The non-trivial Galois automorphism of K{Q p will be denoted by a Þ Ñ ā.Lemma 9.3.There is an isomorphism of K-algebras ψ : D 0 π " Ý Ñ M 2 pKq, ψpaq " ˆa 0 0 ā˙f or all a P W pF q q, ψpF q " ˆ0 1 ip 0 ˙and ψpV q " ˆ0 ´i p 0 ˙.
The integral Dieudonné ring D π is identified by ψ with the o-subalgebra ψpD π q " "ˆa b c d ˙P M 2 po K q ; p|c and a " d pmod pq * (9.1) of index p 4 in the maximal order M 2 po K q.
Proof.The minimal polynomial of π " ip is P π pxq " x 2 `p2 .As defined in Section §2.1, it follows that h π pF, V q " F `V , and (2.5) implies the description D 0 π " W pF q qtF , V u{F V ´p, F 2 `V 2 q b Zp Q p in terms of generators and relations.A computation shows that ψ is well defined.Since both D 0 π and M 2 pKq are central simple algebras over K of degree 2, the map ψ must be an isomorphism.
It remains to compute the image ψpD π q.The elements V , 1, F and F 2 span D π as a left W pF q q-module.Hence ψpD π q is the set of ψpuV `x ¨1 `vF `yF 2 q " ˆx `yip v ´iu pp´ū `ivq x `ȳip ḟor all x, u, y, v P W pF q q.Now the claim follows from "ˆx `yip x `ȳip ˙; x, y P W pF q q * " "ˆa d ˙; a, d P W pF q q, a " d pmod pq * , "ˆv ´iu ´ū `iv ˙; u, v P W pF q q * " "ˆb c 1 ˙; b, c 1 P W pF q q * .
Via the embedding D π Ď M 2 pKq induced by ψ of Lemma 9.3 we obtain two D π -modules as the first column, respectively the second column, of the matrix description.This results in the following embedding of D π -modules.
Proof.Since we have p M 2 po K q Ď D π Ď M 2 po K q, any D π -lattice Λ can by homothety be brought to a position The choice of Λ corresponds to a D π -stable subgroup of Λ 2 {pΛ 2 " o '2 K {po '2 K " F '2 q on which D π acts through the reduction mod p, i.e. the image of This image consists of "ˆa b 0 d ˙P M 2 pF q q ; a " d* .
Therefore the only nontrivial invariant subspace in Λ 2 {pΛ 2 is the image of Λ 1 .
Let Frob : B π Ñ B ppq π be the F q -isogeny given by the relative (p-)Frobenius morphism.We set T p pB π q " Λ and identify Frob via Dieudonné theory with the inclusion of D π -modules T p pB ppq π q " F Λ Ď Λ " T p pB π q, which has a cokernel of W pF q q-length 1 corresponding to the kernel of Frobenius under the Dieudonné functor for finite group schemes.Up to exchanging B π with its Frobenius twist, we may and will assume that T p pB π q " Λ 2 and T p pB ppq π q " Λ 1 Ď Λ 2 .With these identifications in place, the D π -embedding D π ãÑ Λ 1 ' Λ 2 from (9.2) has cokernel of W pF q q-length 1 and determines an isogeny B ppq π ˆBπ ÝÑ Āπ of degree p with T p p Āπ q " D π .
Proposition 9.5.The abelian variety Āπ constructed above is a reduced π-balanced object in AV π and hence an injective cogenerator.
Proof.By construction, T p p Āπ q " D π is a free D π -module of rank 1.For " p, the R π b Zmodule T p Āπ q is free of rank 2 because R π " Zrips and the localization R π r 1 p s " Zri, 1 p s is a Dedekind ring, hence R π is regular away from p.
Remark 9.6.We may formulate the failure for B π to be an injective cogenerator as follows.There are failures in two steps: (1) There are two non-isomorphic D π lattices Λ 1 and Λ 2 that represent non-isomorphic simple objects in AV π , while the target category of End Fq pB π q-modules has only one isomorphism type of simple objects that those can be mapped to.(2) The reduced π-balanced abelian variety Āπ does a better job because first of all it combines both D π -lattices.But B ppq π ˆBπ also does this, and still Hom Fq p´, B ppq π ˆBπ q is not an anti-equivalence, because otherwise B ppq π ˆBπ was π-locally projective and then the same would apply to the direct factor B π , contradiction.
(3) The decisive improvement of Āπ over the product B ppq π ˆBπ comes from choosing an appropriate sublattice D π in the product Λ 1 ' Λ 2 .Now, the abelian variety Āπ has T p p Āπ q " D π and so is also locally projective at p.The construction of the sublattice can be described in terms of a congruence as follows.Although Λ 1 is not isomorphic to Λ 2 as D π -modules, the Frobenius on F q induces an isomorphism of finite D π -modules This congruence yields a description of D π as a fiber product of D π -modules and in some sense it is this coincidence of a congruence between Λ 1 and Λ 2 as D π -lattices which endows T p p Āπ q and a posteriori Āπ with its remarkable properties.
We conclude the example by computing S π " End Fq p Āπ q.
Proposition 9.7.There is an isomorphism S π " End Fq p Āπ q » "ˆa b c d ˙P M 2 pZrisq ; p|c and a " d pmod pq * .
Proof.This follows by the proof of Theorem 4.9 applied to the isogeny of p-power degree ϕ : B π ˆBπ Frob ˆid ÝÝÝÝÝÑ B ppq π ˆBπ Ñ Āπ .As a consequence of Proposition 9.7 the category AV π is anti-equivalent to the category of modules over the non maximal order S π of M 2 pQpiqq which are finite and free as Z-modules.
Remark 4.2.Since we have canonical product decompositions T pAq " ź λ| T λ pAq, ´resp.T p pAq " ź p|p T p pAq ās R w b Z -modules (resp.as D w -modules), we do not get something different in Definition 4.1 if we replace T pAq and T p pAq by their more local versions T λ pAq and T p pAq.
(1) Two R w -orders O Ď E and O 1 Ď E 1 are called Tate-locally isomorphic if for every prime number (including " p) the rings ObZ and O 1 bZ are isomorphic as R w bZ -algebras.(2) Two R w -orders O, O 1 Ď E are called Tate-locally conjugate if for every prime number (including " p) the rings O b Z and O 1 b Z are conjugate inside E b Q .
Fq pAq b Q such that φpO b Z q φ´1 " End Fq pAq b Z in End Fq pAq b Q .Since the stabilizer of O b Z is open, we may approximate φ by an element ϕ P End Fq pAq which still conjugates O b Z onto End Fq pAq b Z .By primary decomposition of kerpϕq we may factor ϕ : A Ñ A as

ζ
v,w : Zpwq ÝÑ Zpvq by ζ v,w prAsq " rBs where B is the abelian subvariety of A generated by the image of all f : C Ñ A with wpCq Ď v. Proposition 7.1 (5) states that ζ v,w indeed takes values in Zpvq.These maps satisfy the compatibility condition ζ u,w " ζ u,v ζ v,w , for all tuples u Ď v Ď w, hence they define a projective system pZpwq, ζ v,w q translate the bottom row of the previous diagram into the top row of the following diagram due to Tate's theorem.EndpV p pA w qq op ˆź EndpV p pB w qq op ˆź |NEndpV pB w qq